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Multivariable Calculus

Oliver Knill

Math 21a, Fall 2012

These notes contain condensed ”two pages per lecture” notes with essential information

only. Remaining space was filled with problems.

Harvard Multivariable Calculus Math 21a, Fall 2012

Math 21a: Multivariable calculus Oliver Knill, Fall 2012

1: Geometry and Distance

A point in the plane has two coordinates P = (x, y). A point in space is de-termined by three coordinates P = (x, y, z). The signs of the coordinates define 4quadrants in the plane and 8 octants in space. These regions by intersect at theorigin O = (0, 0) or O = (0, 0, 0) and are separated by coordinate axes {y = 0 }and {x = 0 } or coordinate planes {x = 0 }, {y = 0 }, {z = 0 }.

1 Describe the location of the points P = (1, 2, 3), Q = (0, 0,−5), R = (1, 2,−3) in words.Possible Answer: P = (1, 2, 3) is in the positive octant of space, where all coordinates

are positive. The point R = (0, 0,−5) is on the negative z axis. The point S = (1, 2,−3) is

below the xy-plane. When projected onto the xy-plane it is in the first quadrant.

2 Problem. Find the midpointM of P = (1, 2, 5) andQ = (−3, 4, 7). Answer. The midpoint

is obtained by taking the average of each coordinate M = (P +Q)/2 = (−1, 3, 6).

The Euclidean distance between two points P =(x, y, z) and Q = (a, b, c) in space is defined as d(P,Q) =√

(x− a)2 + (y − b)2 + (z − c)2.

This definition of Euclidean distance is motivated by the Pythagorean theorem. 1

3 Find the distance d(P,Q) between the points P = (1, 2, 5) and Q = (−3, 4, 7) and verify thatd(P,M) + d(Q,M) = d(P,Q). Answer: The distance is d(P,Q) =

√42 + 22 + 22 =

√24.

The distance d(P,M) is√22 + 12 + 12 =

√6. The distance d(Q,M) is

√22 + 12 + 12 =

√6.

Indeed d(P,M) + d(M,Q) = d(P,Q).

A circle of radius r centered at P = (a, b) is the collection of points in the planewhich have distance r from P .

A sphere of radius ρ centered at P = (a, b, c) is the collection of points in spacewhich have distance ρ from P . The equation of a sphere is (x−a)2+(y−b)2+(z−c)2 =ρ2.

4 Is the point (3, 4, 5) outside or inside the sphere (x−2)2+(y−6)2+(z−2)2 = 16? Answer:

The distance of the point to the center of the sphere is√1 + 4 + 9 which is smaller than 4

the radius of the sphere. The point is inside.

1It appears in an appendix to ”Geometry” of ”Discours de la methode” from 1637, Rene Descartes (1596-1650). More about Descartes in ”Descartes Secret Notebook” by Amir Aczel.

The completion of the square of an equation x2 + bx + c = 0 is the idea to add(b/2)2 − c on both sides to get (x + b/2)2 = (b/2)2 − c. Solving for x gives the

solution x = −b/2 ±√

(b/2)2 − c.

2

5 Solve 2x2 − 10x + 12 = 0. Answer. The equation is equivalent to x2 + 5x = −6. Adding

(5/2)2 on both sides gives (x+ 5/2)2 = 1/4 so that x = 2 or x = 3.

6 Find the center of the sphere x2 + 5x+ y2 − 2y + z2 = −1. Answer: Complete the squareto get (x+5/2)2 − 25/4+ (y− 1)2 − 1+ z2 = −1 or (x− 5/2)2+ (y− 1)2 + z2 = (5/2)2. We

see a sphere center (5/2, 1, 0) and radius 5/2.

Al-Khwarizai Rene Descartes Distance between spheres

7 Find the set of points P = (x, y, z) in space which satisfy x2 + y2 = 9. Answer: This is acylinder of radius 3 around the z-axes parallel to the y axis.

8 What is x2+ y2 = z2. Answer: this is the set of points for which the distance to the z axesis equal to the distance to the xy-plane. It must be a cone.

9 Find the distances of P = (12, 5, 3) to the xy-plane. Answer: 3. Find the distance ofP = (12, 5, 0) to z axes. Answer: 13.

10 Describe x2 + 2x + y2 − 16y + z2 + 10z + 54 = 0. Answer: Complete the square to get a

sphere (x+ 2)2 + (y − 8)2 + (z + 5)2 = 36 with center (−2, 8,−5) and radius 6.

11 Describe the set xz = 0. Answer: We either must have x = 0 or z = 0. The set is a unionof two coordinate planes.

12 Find an equation for the set of points which have the same distance to (1, 1, 1) and (0, 0, 0).Answer: (x− 1)2 + (y − 1)2 + (z − 1)2 = x2 + y2 + z2 gives −2x+ 1− 2y + 1− 2z + 1 = 0

or 2x+ 2y + 2z = 3. This is the equation of a plane.

13 Find the distance between the spheres x2+(y−12)2+z2 = 1 and (x−3)2+y2+(z−4)2 = 9.

Answer:The distance between the centers is√32 + 42 + 122 = 13. The distance between

the spheres is 13− 3− 1 = 9.

2Due to Al-Khwarizmi (780-850) in ”Compendium on Calculation by Completion and Reduction” The book”The mathematics of Egypt, Mesopotamia,China, India and Islam, a Source book, Ed Victor Katz, containstranslations of some of this work.


2: Vectors and Dot product

Two points P = (a, b, c) and Q = (x, y, z) in space define a vector ~v = 〈x− a, y −b − z − c〉. It points from P to Q and we write also ~v = ~PQ. The real numbersnumbers p, q, r in a vector ~v = 〈v1, v2, v3〉 are called the components of ~v.

Similar definitions hold in two dimensions, where vectors have two components. Vectors can bedrawn everywhere in space but two vectors with the same components are considered equal. 1

The addition of two vectors is ~u + ~v = 〈u1, u2, u3〉 + 〈v1, v2, v3〉 =〈u1+v1, u2+v2, u3+v3〉. The scalar multiple λ~u = λ〈u1, u2, u3〉 = 〈λu1, λu2, λu3〉.The difference ~u− ~v can best be seen as the addition of ~u and (−1) · ~v.

The addition and scalar multiplication of vectors satisfy the laws you know from arithmetic.commutativity ~u+~v = ~v+~u, associativity ~u+(~v+ ~w) = (~u+~v)+ ~w and r ∗ (s∗~v) = (r ∗ s)∗~vas well as distributivity (r+s)~v = ~v(r+s) and r(~v+~w) = r~v+r ~w, where ∗ is scalar multiplication.

The length |~v| of a vector ~v = ~PQ is defined as the distance d(P,Q) from P to Q.A vector of length 1 is called a unit vector.

1 |〈3, 4〉| = 5 and |〈3, 4, 12〉| = 13. Examples of unit vectors are |~i| = |~j| = ~k| = 1 and〈3/5, 4/5〉 and 〈3/13, 4/13, 12/13〉. The only vector of length 0 is the zero vector |~0| = 0.

The dot product of two vectors ~v = 〈a, b, c〉 and ~w = 〈p, q, r〉 is defined as ~v · ~w =ap + bq + cr.

The dot product determines distance and distance determines the dot product.

Proof: Lets write v = ~v in this proof. Using the dot product one can express the length of v as|v| = √

v · v. On the other hand, (v+w) · (v+w) = v · v+w ·w+2(v ·w) allows to solve for v ·w:v · w = (|v + w|2 − |v|2 − |w|2)/2 .

The Cauchy-Schwarz inequality tells |~v · ~w| ≤ |~v||~w|.

Proof. We can assume |w| = 1 after scaling the equation. Now plug in a = v ·w into the equation0 ≤ (v−aw)·(v−aw) to get 0 ≤ (v−(v·w)w)·(v−(v·w)w) = |v|2+(v·w)2−2(v·w)2 = |v|2−(v·w)2which means (v · w)2 ≤ |v|2.

The Cauchy-Schwarz inequality allows us to define whatan ”angle” is.

The angle between two nonzero vectors is defined as theunique α ∈ [0, π] which satisfies ~v · ~w = |~v| · |~w| cos(α).

1We cover 2300 years of math from Pythagoras (500 BC), Al Kashi (1400), Cauchy (1800) to Hamilton (1850).

Al Kashi’s theorem: A triangle ABC with side lengths a, b, c and angle α oppositeto c satisfies a2 + b2 = c2 + 2ab cos(α).

Proof. Define ~v = ~AB, ~w = ~AC. Because c2 = |~v− ~w|2 = (~v− ~w) · (~v− ~w) = |~v|2 + |~w|2 − 2~v · ~w,We know ~v · ~w = |~v| · |~w| cos(α) so that c2 = |~v|2 + |~w|2 − 2|~v| · |~w| cos(α) = a2 + b2 − 2ab cos(α).

The triangle inequality tells |~u+ ~v| ≤ |~u|+ |~v|

Proof: |~u+~v|2 = (~u+~v)·(~u+~v) = ~u2+~v2+2~u·~v ≤ ~u2+~v2+2|~u·~v| ≤ ~u2+~v2+2|~u|·|~v| = (|~u|+|~v|)2.

Two vectors are called orthogonal or perpendicular if ~v · ~w = 0. The zero vector~0 is orthogonal to any vector. For example, ~v = 〈2, 3〉 is orthogonal to ~w = 〈−3, 2〉.

Pythagoras theorem: if ~v and ~w are orthogonal, then |v − w|2 = |v|2 + |w|2.

Proof: (~v − ~w) · (~v − ~w) = ~v · ~v + ~w · ~w + 2~v · ~w = ~v · ~v + ~w · ~w. Quod erat demonstrandum. 2

The vector P(~v) = ~v·~w|~w|2

~w is called the projection of ~v

onto ~w. The scalar projection ~v·~w|~w|

is plus or minus

the length of the projection of ~v onto ~w. The vector~b = ~v − P (~v) is a vector orthogonal to ~w.

2 Find the projection of ~v = 〈0,−1, 1〉 onto ~w = 〈1,−1, 0〉. Ansser: P(~v) = 〈1/2,−1/2, 0〉.

3 A wind force ~F = 〈2, 3, 1〉 is applied to a car which drives in the direction of the vector

~w = 〈1, 1, 0〉. Find the projection of ~F onto ~w, the force which accelerates or slows downthe car. Answer: ~w(~F · ~w/|~w|2) = 〈5/2, 5/2, 0〉.

4 How can we visualize the dot product? Answer: Difficult task but lets try. The absolute

value of the dot product is the length of the projection. The dot product is positive if ~v and~w form an acute angle, negative if that angle is obtuse.

5 Given ~v = 〈2, 1, 2〉 and ~w = 〈3, 4, 0〉. Find a vector which is in the plane defined by ~v and ~wand which bisects the angle between these two vectors. Answer. Normalize the two vectors

to make them unit vectors then add them to get 〈13, 17, 10〉/15.

6 Given two vectors ~v, ~w which are perpendicular. Under which condition is ~v+~w perpendicularto ~v − ~w? Answer: Find the dot product of ~v + ~w with ~v − ~w and set it zero.

7 Is the angle between 〈1, 2, 3〉 and 〈−15, 2, 4〉 acute or obtuse? Answer: the dot product is1. Cute!

2Short QED. We have just proven Pythagoras and AlKashi. Distance and angle were defined, not deduced.


3: Cross product

The cross product of two vectors ~v = 〈v1, v2〉 and ~w = 〈w1, w2〉 in the plane is the

scalar v1w2 − v2w1 = det

[

v1 v2w1 w2

]

.

The cross product of two vectors ~v = 〈v1, v2, v3〉 and ~w = 〈w1, w2, w3〉 in space isdefined as the vector

~v × ~w = 〈v2w3 − v3w2, v3w1 − v1w3, v1w2 − v2w1〉 .

1

To remember it we write the product as a ”determinant”:

i j kv1 v2 v3w1 w2 w3

=

iv2 v3w2 w3

−

jv1 v3w1 w3

+

kv1 v2w1 w2

which is ~i(v2w3 − v3w2)−~j(v1w3 − v3w1) + ~k(v1w2 − v2w1).

1 The cross product of 〈1, 2〉 and 〈4, 5〉 is a scalar and 5− 8 = −3.

2 The cross product of 〈1, 2, 3〉 and 〈4, 5, 1〉 is the vector 〈−13, 11,−3〉.

The cross product ~v × ~w is anti-commutative. The re-sulting vector is orthogonal to both ~v and ~w.

Proof. We verify for example that ~v · (~v × ~w) = 0 andlook at the definition.

The sin formula: |~v × ~w| = |~v||~w| sin(α).

Proof: We verify the Lagrange’s identity |~v × ~w|2 = |~v|2|~w|2 − (~v · ~w)2 by direct computation.Now, |~v · ~w| = |~v||~w| cos(α).

The absolute value respectively length |~v × ~w| defines the area of the parallelo-

gram spanned by ~v and ~w.

1It was Hamilton who described in 1843 first a multiplication ∗ of 4 vectors. It contains intrinsically both dotand cross product because (0, v1, v2, v3) ∗ (0, w1, w2, w3) = (−vw, v × w).

The definition shot fits with our intuition: |~w| sin(α) is the height of the parallelogram with baselength |~v|. The area does not change if we rotate the vectors around in space because both lengthand angle stay the same. Area also is linear in each of the vectors v and w. If we make v twice aslong, then the area gets twice as large.

~v × ~w is zero exactly if ~v and ~w are parallel, that is if ~v = λ~w for some real λ.

Proof. This follows immediately from the sin formula and the fact that sin(α) = 0 if α = 0 orα = π.

The cross product can therefore be used to check whether two vectors are parallel or not. Notethat v and −v are also considered parallel even so sometimes one calls this anti-parallel.

The trigonometric sin-formula: if a, b, c are the side lengths of a triangle andα, β, γ are the angles opposite to a, b, c then a/ sin(α) = b/ sin(β) = c/ sin(γ.

Proof. The area of the triangle is ab sin(γ) = bc sin(α) = ac sin(β) Divide the first equation bysin(γ) sin(α) to get one identity. Divide the second equation by sin(α) sin(β) to get the secondidentity.

3 If ~v = 〈a, 0, 0〉 and ~w = 〈b cos(α), b sin(α), 0〉, then ~v× ~w = 〈0, 0, ab sin(α)〉 which has length|ab sin(α)|.

The scalar [~u,~v, ~w] = ~u · (~v × ~w) is called the triple

scalar product of ~u,~v, ~w. The number |[~u,~v, ~w]| de-fines the volume of the parallelepiped spanned by~u,~v, ~w and the orientation of three vectors is the signof [~u,~v, ~w].

These definitions fit intuition: the value h = |~u · ~n|/|~n| is the height of the parallelepiped if~n = (~v × ~w) is a normal vector to the ground parallelogram of area A = |~n| = |~v × ~w|. Thevolume of the parallelepiped is hA = (~u · ~n/|~n|)|~v × ~w| which simplifies to ~u · ~n = |(~u · (~v × ~w)|which is indeed the absolute value of the triple scalar product. The vectors ~v, ~w and ~v × ~w forma right handed coordinate system. If the first vector ~v is your thumb, the second vector ~w isthe pointing finger then ~v × ~w is the third middle finger of the right hand.

4 Problem: Find the volume of a cuboid of width a length b and height c. Answer. The

cuboid is a parallelepiped spanned by 〈a, 0, 0〉〈0, b, 0〉 and 〈0, 0, c〉. The triple scalar productis abc.

5 Problem Find the volume of the parallelepiped which has the vertices O = (1, 1, 0), P =(2, 3, 1), Q = (4, 3, 1), R = (1, 4, 1). Answer: We first see that it is spanned by the vectors

~u = 〈1, 2, 1〉, ~v = 〈3, 2, 1〉, and ~w = 〈0, 3, 1〉. We get ~v× ~w = 〈−1,−3, 9〉 and ~u · (~v× ~w) = 2.The volume is 2.

6 Problem: find the equation ax + by + cz = d for the plane which contains the pointP = (1, 2, 3) as well as the line which passes through Q = (3, 4, 4) and R = (1, 1, 2). To

do so find a vector ~n = 〈a, b, c〉 normal to the and noting (~x − ~OP ) · ~n = 0. Answer: Anormal vector ~n = 〈1,−2, 2〉 = 〈a, b, c〉 of the plane ax+ by+ cz = d is obtained as the cross

product of ~PQ and ~RQ With d = ~n · ~OP = 〈1,−2, 2〉 · 〈1, 2, 3〉 = 3, we get the equationx− 2y + 2z = 3.


4: Lines and Planes

A point P = (p, q, r) and a vector ~v = 〈a, b, c〉 define the line

L = {〈p, q, r〉+ t〈a, b, c〉, t ∈ R } .

The line is obtained by adding a multiple of the vector ~v to the vector ~OP = 〈p, q, r〉. Everyvector contained in the line is necessarily parallel to ~v. We think about the parameter t as ”time”.For t = 0, we are at P and for t = 1 we are at ~OP + ~v.

If t is restricted to the parameter interval [s, u], then L = {〈p, q, r〉+ t〈a, b, c〉, s ≤t ≤ u } is a line segment connecting ~r(s) with ~r(u).

1 Problem. Get the line through P = (1, 1, 2) and Q = (2, 4, 6). Solution. with ~v = ~PQ =〈1, 3, 4〉 we get get L = {〈x, y, z〉 = 〈1, 1, 2〉+ t〈1, 3, 4〉; } which is ~r(t) = 〈1+ t, 1+3t, 2+4t〉.Since 〈x, y, z〉 = 〈1, 1, 2〉+t〈1, 3, 4〉 consists of three equations x = 1+2t, y = 1+3t, z = 2+4twe can solve each for t to get t = (x− 1)/2 = (y − 1)/3 = (z − 2)/4.

The line ~r = ~OP + t~v defined by P = (p, q, r) and vector ~v = 〈a, b, c〉 with nonzeroa, b, c satisfies the symmetric equations

x− p

a=

y − q

b=

z − r

c.

Proof. Each of these expressions is equal to t. These symmetric equations have to be modified abit one or two of the numbers a, b, c are zero. If a = 0, replace the first equation with x = p, ifb = 0 replace the second equation with y = q and if c = 0 replace third equation with z = r.

2 Find the symmetric equations for the line through the two points P = (0, 1, 1) and Q =(2, 3, 4) Solution. first first form the parametric equations 〈x, y, z〉 = 〈0, 1, 1〉+ t〈2, 2, 3〉or x = 2t, y = 1 + 2t, z = 1 + 3t and solve for t to get x/2 = (y − 1)/2 = (z − 1)/3.

3 Problem: Find the symmetric equation for the z axes. Answer: This is a situation where

a = b = 0 and c = 1. The symmetric equations are simply x = 0, y = 0. If two of thenumbers a, b, c are zero, we have a coordinate plane. If one of the numbers are zero, then

the line is contained in a coordinate plane.

A point P and two vectors ~v, ~w define a plane Σ = { ~OP + t~v + s~w, where t, s arereal numbers }.

4 An example is Σ = {〈x, y, z〉 = 〈1, 1, 2〉+ t〈2, 4, 6〉+ s〈1, 0,−1〉 }. This is called the para-

metric description of a plane.

If a plane contains the two vectors ~v and ~w, then the vector ~n = ~v × ~w is orthogonal to both ~vand ~w. Because also the vector ~PQ = ~OQ− ~OP is perpendicular to ~n, we have (Q− P ) · ~n = 0.With Q = (x0, y0, z0), P = (x, y, z), and ~n = 〈a, b, c〉, this means ax+ by + cz = ax0by0 + cz0 = d.The plane is therefore described by a single equation ax+ by + cz = d. We have just shown

The equation of the plane ~x = ~x0 + t~v + s~w

ax+ by + cz = d ,

where 〈a, b, c〉 = ~v × ~w and d is obtained by plugging in ~x0.

5 Problem: Find the equation of a plane which contains the three points P = (−1,−1, 1), Q =

(0, 1, 1), R = (1, 1, 3).Answer: The plane contains the two vectors ~v = 〈1, 2, 0〉 and ~w = 〈2, 2, 2〉. We have

~n = 〈4,−2,−2〉 and the equation is 4x − 2y − 2z = d. The constant d is obtained byplugging in the coordinates of a point to the left. In our case, it is 4x− 2y − 2z = −4.

6 Problem: Find the angle between the planes x+ y = −1 and x+ y+ z = 2. Answer: find

the angle between ~n = 〈1, 1, 0〉 and ~m = 〈1, 1, 1〉. It is arccos(2/√6).

Finally, lets look at some distance functions.

The distance between P and Σ : ~n · ~x = d containing Q is d(P,Σ) = | ~PQ·~n||~n|

.

Proof. Project PQ onto ~n.

The distance between P and the line L is d(P, L) = |( ~PQ)×~u||~u|

.

Proof: the area of the parallelogram spanned by PQ and ~u divided by the base length |~u|.

The lines L : ~r(t) = Q+ t~u,M : ~s(t) = P + t~v have distance d(L,M) = |( ~PQ)·(~u×~v)||~u×~v|

.

Proof. Project PQ onto ~n = ~u× ~v.

The distance between two planes ~n · ~x = d and ~n · ~x = e is d(Σ,Π) = |e−d||~n|

.

Proof: If P is on the first and Q on the second plane, the distance is the scalar projection of ~PQonto ~n. Note that ~PQ · ~n = d− e.

7 A regular tetrahedron has vertices at the points P1 =(0, 0, 3),P2 = (0,

√8,−1), P3 = (−

√6,−

√2,−1) and

P4 = (√6,−

√2,−1). Find the distance between two

edges which do not intersect.


Lecture 5: Functions

A function of two variables f(x, y) is a rule whichassigns to two numbers x, y a third number f(x, y). Forexample, the function f(x, y) = x2y+2x assigns to (3, 2)the number 322 + 6 = 24. The domain D of a func-tion is set of points where f is defined, the range is{f(x, y) | (x, y) ∈ D }. The graph of f(x, y) is the sur-face {(x, y, f(x, y)) | (x, y) ∈ D } in space Graphs allowto visualize functions.

1 The graph of f(x, y) =√

1− (x2 + y2) on the domain D = {x2 + y2 < 1 } is a half sphere.The range is the interval [0, 1].

The set f(x, y) = c = const is called a contour curve or level curve of f . Forexample, for f(x, y) = 4x2 + 3y2, the level curves f = c are ellipses if c > 0. Thecollection of all contour curves {f(x, y) = c } is called the contour map of f .

2 For f(x, y) = x2 − y2, the set x2 − y2 = 0 is the union of the lines x = y and x = −y. Thecurve x2 − y2 = 1 is made of two hyperbola with with their ”noses” at the point (−1, 0)

and (1, 0). The curve x2 − y2 = −1 consists of two hyperbola with their noses at (0, 1) and

(0,−1).

3 For f(x, y) = (x2 − y2)e−x2−y

2

, we cannot find explicit expressions for the con-

tour curves (x2 − y2)e−x2−y

2

= c. butwe can draw the curves with the com-

puter:

A function of three variables g(x, y, z) assigns to three variables x, y, z a real num-ber g(x, y, z). We can visualize it by contour surfaces g(x, y, z) = c, where c isconstant. It is helpful to look at the traces, the intersections of the surfaces withthe coordinate planes x = 0, y = 0 or z = 0.

4 For g(x, y, z) = z−f(x, y), the level surface g = 0 which is the graph z = f(x, y) of a functionof two variables. For example, for g(x, y, z) = z−x2−y2 = 0, we have the graph z = x2+y2

of the function f(x, y) = x2 + y2 which is a paraboloid. Most surfaces g(x, y, z) = c are notgraphs.

5 If f(x, y, z) is a polynomial and f(x, x, x) is quadratic in x, then {f = c} is a quadric.

Sphere Paraboloid Plane

x2 + y2 + z2 = 1 x2 + y2 − c = z ax+ by + cz = d

One sheeted Hyperboloid Cylinder Two sheeted Hyperboloid

x2 + y2 − z2 = 1 x2 + y2 = r2 x2 + y2 − z2 = −1

Ellipsoid Hyperbolic paraboloid Elliptic hyperboloid

x2/a2 + y2/b2 + z2/c2 = 1 x2 − y2 + z = 1 x2/a2 + y2/b2 − z2/c2 = 1


Lecture 6: Curves

A parametrization of a planar curve is a map ~r(t) = 〈x(t), y(t)〉 from a parameter

interval R = [a, b] to the plane. The functions x(t), y(t) are called coordinate

functions. The image of the parametrization is called a parametrized curve inthe plane. The parametrization of a space curve is ~r(t) = 〈x(t), y(t), z(t)〉. Theimage of r is a parametrized curve in space.

rHt L

We always think of the parameter t as time. For a fixed t, we have a vector 〈x(t), y(t), z(t)〉 inspace. As t varies, the end point of this vector moves along a curve. The parametrization containsmore information about the curve then the curve. It tells also how fast and in which direction wetrace the curve.

1 The parametrization ~r(t) = 〈1 + 3 cos(t), 3 sin(t)〉 is a circle of radius 3 centered at (1, 0)

2 ~r(t) = 〈cos(3t), sin(5t)〉 defines a Lissajous curve example.

3 If x(t) = t, y(t) = f(t), the curve ~r(t) = 〈t, f(t)〉 traces the graph of the function f(x). For

example, for f(x) = x2 + 1, the graph is a parabola.

4 With x(t) = 2 cos(t), y(t) = sin(t), then ~r(t) follows an ellipse x(t)2/4 + y(t)2 = 1.

5 The space curve ~r(t) = 〈t cos(t), t sin(t), t〉 traces a helix with increasing radius.

6 If x(t) = cos(2t), y(t) = sin(2t), z(t) = 2t is the same curve as before but the parameteri-

zation has changed.

7 With x(t) = cos(−t), y(t) = sin(−t), z(t) = −t it is traced in the opposite direction.

8 With ~r(t) = 〈cos(t), sin(t)〉 + 0.1〈cos(17t, sin(17t)〉 we have an example of an epicycle,

where a circle turns on a circle. It was used in the Ptolemaic geocentric system whichpredated the Copernican system still using circular orbits and then the modern Keplerian

system, where planets move on ellipses and which can be derived from Newton’s laws.

If ~r(t) = 〈x(t), y(t), z(t)〉 is a curve, then ~r ′(t) = 〈x′(t), y′(t), z′(t)〉 = 〈x, y, z〉 iscalled the velocity at time t. Its length |~r′(t)| is called speed and ~v/|~v| is calleddirection of motion. The vector ~r ′′(t) is called the acceleration. The thirdderivative ~r ′′′ is called the jerk.

Any vector parallel to ~r ′(t) is called tangent to the curve at ~r(t).

The addition rule in one dimension (f+g)′ = f ′+g′, the scalar multiplication rule (cf)′ = cf ′

and the Leibniz rule (fg)′ = f ′g+fg′ and the chain rule (f(g))′ = f ′(g)g′ generalize to vector-valued functions because in each component, we have the single variable rule.The process of differentiation of a curve can be reversed using the fundamental theorem of

calculus. If ~r′(t) and ~r(0) is known, we can figure out ~r(t) by integration ~r(t) = ~r(0)+∫t

0~r′(s) ds.

Assume we know the acceleration ~a(t) = ~r′′(t) at all times as well as initial velocity

and position ~r′(0) and ~r(0). Then ~r(t) = ~r(0)+t~r′(0)+ ~R(t), where ~R(t) =∫t

0~v(s) ds

and ~v(t) =∫t

0~a(s) ds.

The free fall is the case when acceleration is constant. In particular, if ~r′′(t) = 〈0, 0,−10〉,~r′(0) = 〈0, 1000, 2〉, ~r(0) = 〈0, 0, h〉, then ~r(t) = 〈0, 1000t, h+ 2t− 10t2/2〉.

If r′′(t) = ~F is constant, then ~r(t) = ~r(0) + t~r′(0)− ~F t2/2.

50 100 150 200 250

-2

2

4

6

8

10


7: Arc length and curvature

If t ∈ [a, b] 7→ ~r(t) is a curve with velocity ~r ′(t) and speed |~r ′(t)|, then L =∫ ba |~r ′(t)| dt is called the arc length of the curve. In space the length is L =∫ ba

√

x′(t)2 + y′(t)2 + z′(t)2 dt.

1 The arc length of the circle of radius R given by ~r(t) = 〈R cos(t), R sin(t)〉 parameterizedby 0 ≤ t ≤ 2π is 2π because the speed |~r′(t)| is constant and equal to R. The answer is

2πR.

2 The helix ~r(t) = 〈cos(t), sin(t), t〉 has velocity ~r ′(t) = 〈− sin(t), cos(t), 1〉 and constant

speed |~r ′(t)| = 〈− sin(t), cos(t), 1〉 =√2.

3 What is the arc length of the curve

~r(t) = 〈t, log(t), t2/2〉

for 1 ≤ t ≤ 2? Answer: Because ~r′(t) = 〈1, 1/t, t〉, we have ~r′(t) =√

1 + 1t2+ t2 = |1

t+ t|

and L =∫ 21

1t+ t dt = log(t) + t2

2|21 = log(2) + 2 − 1/2. Because the curve constructed in

such a way that the arc length can be computed, we call it ”opportunity”.

4 Find the arc length of the curve ~r(t) = 〈3t2, 6t, t3〉 from t = 1 to t = 3.

5 What is the arc length of the curve ~r(t) = 〈cos3(t), sin3(t)〉, 0 ≤ t ≤ 2π? Answer: We have

|~r′(t)| = 3√

sin2(t) cos4(t) + cos2(t) sin4(t) = (3/2)| sin(2t)|. Therefore, ∫ 2π0 (3/2) sin(2t) dt =

6.

6 Find the arc length of ~r(t) = 〈t2/2, t3/3〉 for −1 ≤ t ≤ 1. This cubic curve satisfies

y2 = x38/9 and is an example of an elliptic curve. The speed is |~r(t)| =√t2 + t4. Because

∫

x√1 + x2 dx = (1+x2)3/2/3, the arc length integral can be evaluated as

∫ 1−1 |t|

√1 + t2 dx =

2∫ 10 t

√1 + t2 dt = 2(1 + t2)3/2/3|10 = 2(2

√2− 1)/3.

7 The arc length of an epicycle ~r(t) = 〈t + sin(t), cos(t)〉 parameterized by 0 ≤ t ≤ 2π. We

have |~r′(t)| =√

2 + 2 cos(t). so that L =∫ 2π0

√

2 + 2 cos(t) dt. A substitution t = 2u

gives L =∫ π0

√

2 + 2 cos(2u) 2du =∫ π0

√

2 + 2 cos2(u)− 2 sin2(u) 2du =∫ π0

√

4 cos2(u) 2du =4∫ π0 | cos(u)| du = 8.

8 Find the arc length of the catenary ~r(t) = 〈t, cosh(t)〉, where cosh(t) = (et + e−t)/2 is the

hyperbolic cosine and t ∈ [−1, 1]. We have

cosh2(t)2 − sinh2(t) = 1 ,

where sinh(t) = (et − e−t)/2 is the hyperbolic sine. The answer is∫ 1−1 cosh(t) dt =

2 sinh(1).

Because a parameter change t = t(s) corresponds to a substitution in the integration which doesnot change the integral, we immediately have

The arc length is independent of the parameterization of the curve.

9 The circle parameterized by ~r(t) = 〈cos(t2), sin(t2)〉 on t = [0,√2π] has the velocity ~r ′(t) =

2t(− sin(t), cos(t)) and speed 2t. The arc length is still∫

√2π

0 2t dt = t2|√2π

0 = 2π.

10 We do not always have a closed formula for the arc length of a curve. The length of the

Lissajous figure ~r(t) = 〈cos(3t), sin(5t)〉 leads to∫ 2π0

√

9 sin2(3t) + 25 cos2(5t) dt which

needs to be evaluated numerically.

Define the unit tangent vector ~T (t) = ~r ′(t)|/|~r ′(t)| unit tangent vector.

The curvature if a curve at the point ~r(t) is defined as κ(t) = |~T ′(t)||~r ′(t)| .

The curvature is the magnitude of the acceleration vector if ~r(t) traces the curve with constantspeed 1. A large curvature at a point means that the curve turns sharply. Unlike the accelerationor the velocity, the curvature does not depend on the parameterization of the curve. You ”see”the curvature, while you ”feel” the acceleration.

The curvature does not depend on the parametrization.

Proof. If s(t) be an other parametrization, then by the chain rule d/dtT ′(s(t)) = T ′(s(t))s′(t) andd/dtr(s(t)) = r′(s(t))s′(t). We see that the s′ cancels in T ′/r′.

Especially, if the curve is parametrized by arc length, meaning that the velocity vector r′(t) haslength 1, then κ(t) = |T ′(t)|. It measures the rate of change of the unit tangent vector.

11 The curve ~r(t) = 〈t, f(t)〉, which is the graph of a function f has the velocity ~r ′(t) = (1, f ′(t))

and the unit tangent vector ~T (t) = (1, f ′(t))/√

1 + f ′(t)2. After some simplification we get

κ(t) = |~T ′(t)|/|~r ′(t)| = |f ′′(t)|/√

1 + f ′(t)23

For example, for f(t) = sin(t), then κ(t) = | sin(t)|/|√

1 + cos2(t)3.

If ~r(t) is a curve which has nonzero speed at t, then we can define ~T (t) = ~r ′(t)|~r ′(t)| , the

unit tangent vector, ~N(t) =~T ′(t)

|~T ′(t)| , the normal vector and ~B(t) = ~T (t) × ~N(t)

the bi-normal vector. The plane spanned by ~N and ~B is called the normal

plane. It is perpendicular to the curve. The plane spanned by T and N is calledthe osculating plane.

If we differentiate ~T (t) · ~T (t) = 1, we get ~T ′(t) · ~T (t) = 0 and see that ~N(t) is perpendicular to~T (t). Because B is automatically normal to T and N , we have shown:

The three vectors (~T (t), ~N(t), ~B(t)) are unit vectors orthogonal to each other.

A useful formula for curvature is

κ(t) =|~r ′(t)× ~r ′′(t)|

|~r ′(t)|3


8: Polar and spherical coordinates

A point (x, y) in the plane has the polar coordinates

r =√x2 + y2, θ = arctg(y/x). We have the relation

(x, y) = (r cos(θ), r sin(θ)).

Θ

r

x

y

The formula θ = arctg(y/x) defines the angle θ only up to an addition of π. The points (x, y) and(−x,−y) have the same θ value. To get the correct θ, one can choose arctan(y/x) in (−π/2, π/2],where π/2 when (x, y) is on the positive y axes, and add π on the left plane including the negativey axes.

A curve given in polar coordinates as r(θ) = f(θ) iscalled a polar curve. It can in Cartesian coordinatesbe described as ~r(t) = 〈f(t) cos(t), f(t) sin(t)〉.

1 Describe the curve r = θ in Cartesian coordinates. Solution A formal substitution gives√x2 + y2 = arctan(y/x) but we can do better. Remember that for the curve r(t) =

〈r cos(t), r sin(t) we have exactly the relation r = t. The curve is a spiral.

2 What is the curve r = |2 sin(θ)|? Solution Lets ignore the absolute value for a moment andlook at r2 = 2r sin(θ). This can be written as x2 + y2 = 2y which is x2 + y2 − 2y + 1 = 1.

The curve is a circle of radius 1 centered at (0, 1). Since we have the absolute value, the

radius at θ and θ + π is the same and add the circle of radius 1 centered at (0,−1).

If we represent points in space as

(x, y, z) = (r cos(θ), r sin(θ), z)

we speak of cylindrical coordinates.

Here are some surfaces described in cylindrical coordinates:

3 r = 1 is a cylinder,

4 r = |z| is a double cone

5 θ = 0 is a half plane

6 r = θ is a rolled sheet of paper

7 r = 2 + sin(z) is an example of a surface of revolution.

Spherical coordinates use the distance ρ to the origin as well as two angles θ andφ. The first angle θ is the polar angle in polar coordinates of the xy coordinates andφ is the angle between the vector ~OP and the z-axis. The relation is

(x, y, z) = (ρ cos(θ) sin(φ), ρ sin(θ) sin(φ), ρ cos(φ)) .

There are two important figures to see the connection. The distance to the z axes r = ρ sin(φ)and the height z = ρ cos(φ) can be read off by the left picture the rz-plane, the coordinatesx = r cos(θ), y = r sin(θ) can be seen in the right picture the xy-plane.

x

y

Φ Ρr x = ρ cos(θ) sin(φ),

y = ρ sin(θ) sin(φ),z = ρ cos(φ)

x

y

Θr

Here are some level surfaces described in spherical coordinates:

8 ρ = 1 is a sphere,

9 The surface φ = π/2 is a single cone

10 The surface sin(θ) = cos(φ) is a plane.

11 ρ = φ is an apple shaped surface

12 ρ = 2 + cos(3θ) sin(φ) is an example of a bumpy sphere.

13 Write x2 + y2 − 5x = z2 in cylindrical coordinates! Answer: r2 − 5r cos(θ) = z2.

14 Match the surfaces with ρ = | sin(3φ)|, ρ = | sin(3θ)| in spherical coordinates (ρ, θ, φ). It

helps to see this in the rz plane or the xy plane.


9: Parametrized surfaces

Beside an implicit equation g(x, y, z) = 0, the parametrization is a second, fundamentallydifferent way to describe a surface.

A parametrization of a surface is a vector-valued function

~r(u, v) = 〈x(u, v), y(u, v), z(u, v)〉 ,

where x(u, v), y(u, v), z(u, v) are three functions of two variables.

Because two parameters u and v are involved, the map ~r from the plane to space is also calleduv-map.

A parametrized surface is the image of the uv-map.The domain of the uv-map is called the parameter do-

main.

If the first parameter u is kept constant, then v 7→ ~r(u, v) is a curve on the surface.Similarly, for constant v, the map u 7→ ~r(u, v) traces a curve on the surface. Thesecurves are called grid curves.

A computer draws surfaces using grid curves. The world of parametric surfaces is intriguing. Itcan be explored with the help of a computer. Keep in mind the following 4 important examples.They cover a wide range of cases.

I Planes. Parametric: ~r(s, t) = ~OP + s~v + t~wImplicit: ax + by + cz = d. Parametric to Implicit: find thenormal vector ~n = ~v × ~w.Implicit to Parametric: find two vectors ~v, ~w normal to thevector ~n. For example, find three points P,Q,R on the surfaceand forming ~u = ~PQ,~v = ~PR.

II Spheres: Parametric: ~r(u, v) = 〈a, b, c〉 +〈ρ cos(u) sin(v), ρ sin(u) sin(v), ρ cos(v)〉.Implicit: (x− a)2 + (y − b)2 + (z − c)2 = ρ2.Parametric to implicit: read off the radius and the centerImplicit to parametric: find the center (a, b, c) and the radius rpossibly by completing the square.

III Graphs:Parametric: ~r(u, v) = 〈u, v, f(u, v)〉Implicit: z − f(x, y) = 0. Parametric to implicit: think aboutz = f(x, y)Implicit to parametric: use x and y as the parameterizations.

IV Surfaces of revolution:Parametric: ~r(u, v) = 〈g(v) cos(u), g(v) sin(u), v〉Implicit:

√x2 + y2 = r = g(z) can be written as x2 + y2 = g(z)2.

Parametric to implicit: read off the function g(z) the distance tothe z-axis.Implicit to parametric: again, the function g is the key link.

1 Describe the surface ~r(u, v) = 〈v5 cos(u), v5 sin(u), v〉, where u ∈ [0, 2π] and v ∈ R. Solu-

tion. It is a surface of revolution. We have r = v5 = z5. Draw this in the rz-plane. You

see that r is small the origin making the surface pointy like a needle at the tip.

2 Find a parametrization for the plane which contains the three points P = (3, 7, 1),Q =(6, 2, 1) and R = (0, 3, 4). Solution. Take ~r(s, t) = ~OP + s ~QP + t ~RP . ~r(s, t) = (3 − 3s−3t, 7− 5s− 4t, 1 + 3t).

3 Parametrize the lower half of the ellipsoid x2/4 + y2/9 + z2/25 = 1, z < 0. Solution. One

solution is to solve for z and write it as a graph ~r(u, v) = 〈u, v,−√

25− 25u2/4− 25v2/9〉.We can also deform a sphere ~r(θ, φ) = 〈2 sin(φ) cos(θ), 3 sin(φ) sin(θ), 5 cos(φ)〉.

4 Parametrize the upper half of the hyperboloid x2 + y2/4− z2 = −1. Solution. The roundhyperboloid, where r2 = z2 − 1 is given by ~r(θ, z) = 〈

√z2 − 1 cos(θ),

√z2 − 1 sin(θ), z〉.

Deform this now to get ~r(θ, z) = 〈√z2 − 1 cos(θ), 2

√z2 − 1 sin(θ), z〉.

5

Describe the surface ~r(u, v) = 〈2 + sin(13u) +

sin(17v))〈cos(u) sin(v), sin(u) sin(v), cos(v)〉. Solution.

It looks a bit like a hedgehog. It is an example of a

”bumpy sphere”. The radius ρ is a function of theangles. In spherical coordinates, we have ρ = (2 +

sin(13θ) + sin(17φ)).


10: Continuity

A function f(x, y) with domain R is is continuous at a point (a, b) ∈ R iff(x, y) → f(a, b) whenever (x, y) → (a, b). The function f is continuous on

R, if f is continuous for every point (a, b) on R.

1 a) f(x, y) = x2 + y4 + xy + sin(y + sin sin sin sin(x)2 is continuous on the entire plane. It isbuilt up from functions which are continuous using addition, multiplication or composition

of functions which are all continuous everywhere.

2 f(x, y) = 1/(x2+y2) is continuous everywhere except at the origin, where it is not defined.

3 f(x, y) = y+ sin(x)/|x| is continuous away from x = 0. At every point (0, y) it is discontin-

uous. f(1/n, y) → y + 1 and f(−1/n, y) → y − 1 for n → ∞.

4 f(x, y) = sin(1/(x+ y)) is continuous except on the line x+ y = 0.

5 f(x, y) = (x4 − y4)/(x2 + y2) is continuous at (0, 0). You can divide out x2 + y2 to see thatthe function is equivalent to x2 − y2 away from (0, 0). After defining f(0, 0) = 0 we see that

the function is continuous.

6 There are three reasons, why a function can be discontinuous: it can jump, it can diverge

to infinity, or it can oscillate. An example of a jump appears with f(x) = sin(x)/|x|, apole g(x) = 1/x leads to a vertical asymptote and the function going to infinity. An exampleof a function discontinuous due to oscillations is h(x) = sin(1/x). Its graph is the devil’s

comb.

The prototypes in one dimensions are

Jump f(x) = sin(x)/|x| -5 5

-1.0

-0.5

0.5

1.0

Diverge g(x) = 1/x -1.0 -0.5 0.5 1.0

-10

-5

0

5

10

Oscillate h(x) = sin(1/x)-1.0 -0.5 0.5 1.0

-1.0

-0.5

0.5

1.0

One can have mixtures of these phenomena like the function 3 sin(x)/|x|+ sin(1/x), which jumpsand also has an oscillatory problem at x = 0.

-1.0 -0.5 0.5 1.0

-4

-2

0

4

There are two handy tools to discover a discontinuities:1) Use polar coordinates with coordinate center at the point to analyze the function.2) Restrict the function to one dimensional curves and check continuity on thatcurve, where one has a function of one variables.

7 Determine whether the function f(x, y) = sin(x2+y2)

x2+y2is continuous at (0, 0). Solution Use

polar coordinates to write this as sin(r2)/r2 which is continuous at 0 (apply l’Hopital twiceif you want to verify this).

8 Is the function f(x, y) = x2−y

2

x2+y2continuous at (0, 0)? Solution Use polar coordinates to see

that this cos(2θ). We see that the value depends on the angle only. Arbitrarily close to(0, 0), the function takes any value from −1 to 1.

9 Is the function f(x, y) = x2y

x4+y2continuous? Solution. Look on the line x2 = y to get the

function x4/(2x4) = 1/(2x2). It is not continuous at 0. This example is a real shocker

because it is continuous through each line through the origin: if y = ax, then f(x, ax) =

ax3/(x4 + a2x2) = ax/(x2 + a2) which is goes to zero for x → 0 as long as a 6= 0. If a = 0however, we have y = 0 and f = 0/x4 which can be continuously extended to x = 0 too.

10 What about the function f(x, y) = xy2+y

3

x2+y2? Solution. Use polar coordinates and write

r3 sin2(θ)(cos(θ) + sin(θ)/r2 = r sin2(θ)(cos(θ) + sin(θ) which shows that the function con-

verges to 0 as r → 0.

11 Is the function f(x, y) = sin(x2+y2)√

x2+y2continuous everywhere? Solution. Use polar coordinates

to see that this is sin(r2)/r2. This function is continuous at 0 by Hopital’s theorem.


11: Partial derivatives

If f(x, y) is a function of two variables, then ∂∂xf(x, y) is defined as the derivative

of the function g(x) = f(x, y), where y is considered a constant. It is called partial

derivative of f with respect to x. The partial derivative with respect to y is definedsimilarly.

We also write fx(x, y) =∂∂xf(x, y). and fyx = ∂

∂x∂∂yf . 1

1 For f(x, y) = x4 − 6x2y2 + y4, we have fx(x, y) = 4x3 − 12xy2, fxx = 12x2 − 12y2, fy(x, y) =

−12x2y+4y3, fyy = −12x2+12y2 and see that fxx+ fyy = 0. A function which satisfies this

equation is also called harmonic. The equation fxx + fyy = 0 is an example of a partial

differential equation for the unknown function f(x, y) involving partial derivatives. The

vector 〈fx, fy〉 is called the gradient.

Clairaut’s theorem If fxy and fyx are both continuous, then fxy = fyx.

Proof: we look at the equations without taking limits first. We extend the definition and say thata background Planck constant h is positive, then fx(x, y) = [f(x + h, y)− f(x, y)]/h. For h = 0we define fx as before. Compare the two sides for fixed h > 0:

hfx(x, y) = f(x+ h, y)− f(x, y)h2fxy(x, y) = f(x+h, y+h)−f(x+h, y+h)− (f(x+ h, y)− f(x, y))

hfy(x, y) = f(x, y + h)− f(x, y).h2fyx(x, y) = f(x + h, y + h) − f(x +h, y)− (f(x, y + h)− f(x, y))

No limits were taken. We established an identity which holds for all h > 0, the discrete derivativesfx, fy satisfy fxy = fyx. It is a ”quantum Clairaut” theorem. If the classical derivatives fxy, fyx areboth continuous, the limit h → 0 leads to the classical Clairaut’s theorem. The quantum Clairauttheorem holds for any functions f(x, y) of two variables. Not even continuity is needed.

2 Find fxxxxxyxxxxx for f(x) = sin(x) + x6y10 cos(y). Answer: Do not compute, but think.

3 The continuity assumption for fxy is necessary. The example f(x, y) = x3y−xy3

x2+y2contradicts

Clairaut’s theorem:

fx(x, y) = (3x2y − y3)/(x2 + y2) − 2x(x3y −xy3)/(x2+y2)2, fx(0, y) = −y, fxy(0, 0) = −1,

fy(x, y) = (x3 − 3xy2)/(x2 + y2) − 2y(x3y −xy3)/(x2 + y2)2, fy(x, 0) = x, fy,x(0, 0) = 1.

An equation for an unknown function f(x, y) which involves partial derivatives withrespect to at least two different variables is called a partial differential equation.If only the derivative with respect to one variable appears, it is called an ordinary

differential equation.

1∂xf, ∂yf were introduced by Carl Gustav Jacobi. Josef Lagrange had used the term ”partial differences”.

Here are some examples of partial differential equations. You should know the first 4 well.

4 Thewave equation ftt(t, x) = fxx(t, x) governs the motion of light or sound. The function

f(t, x) = sin(x− t) + sin(x+ t) satisfies the wave equation.

5 The heat equation ft(t, x) = fxx(t, x) describes diffusion of heat or spread of an epi-

demic. The function f(t, x) = 1√

te−x2/(4t) satisfies the heat equation.

6 The Laplace equation fxx + fyy = 0 determines the shape of a membrane. The function

f(x, y) = x3 − 3xy2 is an example satisfying the Laplace equation.

7 The advection equation ft = fx is used to model transport in a wire. The function

f(t, x) = e−(x+t)2 satisfy the advection equation.

8 The eiconal equation f 2x + f 2

y = 1 is used to see the evolution of wave fronts in optics.

The function f(x, y) =√x2 + y2 satisfies the eiconal equation.

9 The Burgers equation ft + ffx = fxx describes waves at the beach which break. The

function f(t, x) = xt

√1te−x2/(4t)

1+√

1te−x2/(4t)

satisfies the Burgers equation.

10 The KdV equation ft + 6ffx + fxxx = 0 models water waves in a narrow channel.

The function f(t, x) = a2

2cosh−2(a

2(x− a2t)) satisfies the KdV equation.

11 The Schrodinger equation ft =ih2m

fxx is used to describe a quantum particle of mass

m. The function f(t, x) = ei(kx−h2m

k2t) solves the Schrodinger equation. [Here i2 = −1 is

the imaginary i and h is the Planck constant h ∼ 10−34Js.]

Here are the graphs of the solutions of the equations. Can you match them with the PDE’s?


14: Linearization

The linear approximation of a function f(x) at apoint a is the linear function

L(x) = f(a) + f ′(a)(x− a) .

y=LHxL

y=fHxL

The graph of the function L is close to the graph of f near a. We generalize this to higherdimensions:

The linear approximation of f(x, y) at (a, b) is the linear function

L(x, y) = f(a, b) + fx(a, b)(x− a) + fy(a, b)(y − b) .

The linear approximation of a function f(x, y, z) at (a, b, c) is

L(x, y, z) = f(a, b, c) + fx(a, b, c)(x− a) + fy(a, b, c)(y − b) + fz(a, b, c)(z − c) .

Using the gradient ∇f(x, y) = 〈fx, fy〉 rsp. ∇f(x, y, z) = 〈fx, fy, fz〉, the linearization can bewritten as L(~x) = f(~x0)+∇f(~a) ·(~x−~a). By keeping the second variable y = b is fixed, we get to aone-dimensional situation, where the only variable is x. Now f(x, b) = f(a, b)+fx(a, b)(x−a) is thelinear approximation. Similarly, if x = x0 is fixed y is the single variable, then f(x0, y) = f(x0, y0)+fy(x0, y0)(y − y0). Knowing the linear approximations in both the x and y variables, we can getthe general linear approximation by f(x, y) = f(x0, y0) + fx(x0, y0)(x − x0) + fy(x0, y0)(y − y0).Please avoid the notion of differentials. It is a relict from old times.

1 What is the linear approximation of the function f(x, y) = sin(πxy2) at the point (1, 1)? Wehave (fx(x, y), yf(x, y) = (πy2 cos(πxy2), 2yπ cos(πxy2)) which is at the point (1, 1) equal to

∇f(1, 1) = 〈π cos(π), 2π cos(π)〉 = 〈−π, 2π〉.

2 Linearization can be used to estimate functions near a point. In the previous example,

−0.00943 = f(1+0.01, 1+0.01) ∼ L(1+0.01, 1+0.01) = −π0.01−2π0.01+3π = −0.00942 .

3 Find the linear approximation to f(x, y, z) = xy + yz + zx at the point (1, 1, 1). Sincef(1, 1, 1) = 3, and ∇f(x, y, z) = 〈y + z, x + z, y + x〉,∇f(1, 1, 1) = 〈2, 2, 2〉. we have

L(x, y, z) = f(1, 1, 1) + 〈2, 2, 2〉 · 〈x− 1, y − 1, z − 1〉 = 3 + 2(x− 1) + 2(y − 1) + 2(z − 1) =2x+ 2y + 2z − 3.

4 Estimate f(0.01, 24.8, 1.02) for f(x, y, z) = ex√yz.

Solution: take (x0, y0, z0) = (0, 25, 1), where f(x0, y0, z0) = 5. Solution.The gradient

is ∇f(x, y, z) = (ex√yz, exz/(2

√y), ex

√y). At the point (x0, y0, z0) = (0, 25, 1) the gra-

dient is the vector (5, 1/10, 5). The linear approximation is L(x, y, z) = f(x0, y0, z0) +∇f(x0, y0, z0)(x−x0, y−y0, z−z0) = 5+(5, 1/10, 5)(x−0, y−25, z−1) = 5x+y/10+5z−2.5.

We can approximate f(0.01, 24.8, 1.02) by 5+〈5, 1/10, 5〉·〈0.01,−0.2, 0.02〉= 5+0.05−0.02+0.10 = 5.13. The actual value is f(0.01, 24.8, 1.02) = 5.1306, very close to the estimate.

5 Find the tangent line to the graph of the function g(x) = x2 at the point (2, 4). Solution:the tangent line is the level curve of the linearlization of L(x, y) of f(x, y) = y−x2 = 0 which

passes through the point. We compute the gradient 〈a, b〉 = ∇f(2, 4) = 〈−g′(2), 1〉 = 〈−4, 1〉and forming ax + by = −4x + y = d, where d = −4 · 2 + 1 · 4 = −4. The answer is

−4x+ y = −4 .

6 The Barth surface is defined as the level surface f = 0 of

f(x, y, z) = (3 + 5t)(−1 + x2 + y2 + z2)2(−2 + t+ x2 + y2 + z2)2

+ 8(x2 − t4y2)(−(t4x2) + z2)(y2 − t4z2)(x4 − 2x2y2 + y4 − 2x2z2 − 2y2z2 + z4) ,

where t = (√5 + 1)/2 is a constant called the golden ratio. If we replace t with 1/t =

(√5− 1)/2 we see the surface to the middle. For t = 1, we see to the right the surface

f(x, y, z) = 8. Find the tangent plane of the later surface at the point (1, 1, 0). Solution: We

find the level curve of the linearization by computing the gradient ∇f(1, 1, 0) = 〈64, 64, 0〉.The surface is x+ y = d for some constant d. By plugging in the point (1, 1, 0) we see that

x+ y = 2.

7

The quartic surface

f(x, y, z) = x4 − x3 + y2 + z2 = 0

is called the piriform. What is the equation for the tan-gent plane at the point P = (2, 2, 2) of this pair shaped

surface? Solution. We get 〈a, b, c〉 = 〈20, 4, 4〉 and sothe equation of the plane 20x+ 4y+ 4z = 56, where we

have obtained the constant to the right by plugging inthe point (x, y, z) = (2, 2, 2).


15: Chain rule

If f and g are functions of one variable t, then the single variable chain rule tells

d

dtf(g(t)) = f ′(g(t))g′(t) .

For example, d/dt sin(log(t)) = cos(log(t))/t. It can be proven by linearizing the functions f andg and verifying the chain rule in the linear case. The chain rule is useful to find derivatives like

arccos′(x): write 1 = d/dx cos(arccos(x)) = − sin(arccos(x)) arccos′(x) = −√

1− sin2(arccos(x)) arccos′(x)√1− x2 arccos′(x) so that arccos′(x) = −1/

√1− x2.

1 Derive using implicit differentiation the derivative d/dx arctan(x). Solution. We havesin′ = cos and cos′ = − sin and from cos2(x) + sin2(x) = 1. follows 1+ tan2(x) = 1/ cos2(x).

Therefore d/dx tan(arctan(x)) = 1/ cos2(arctan(x)) tan′(x) = x Now 1/ cos2(x) = 1/(1 +tan2(x) so that tan′(x) = 1/(1 + x2).

Define the gradient ∇f(x, y) = 〈fx(x, y), fy(x, y)〉 or ∇f(x, y, z) =〈fx(x, y, z), fy(x, y, z), fz(x, y, z)〉.

If ~r(t) is curve and f is a function of several variables we can build a function t 7→ f(~r(t)) of onevariable. Similarly, If ~r(t) is a parametrization of a curve in the plane and f is a function of twovariables, then t 7→ f(~r(t)) is a function of one variable.

The multivariable chain rule is

d

dtf(~r(t)) = ∇f(~r(t)) · ~r′(t) .

Proof. When written out in two dimensions, it is

d

dtf(x(t), y(t)) = fx(x(t), y(t))x

′(t) + fy(x(t), y(t))y′(t) .

Now, the identity

f(x(t+h),y(t+h))−f(x(t),y(t))h

= f(x(t+h),y(t+h))−f(x(t),y(t+h))h

+ f(x(t),y(t+h))−f(x(t),y(t))h

holds for every h > 0. The left hand side converges to ddtf(x(t), y(t)) in the limit h → 0 and

the right hand side to fx(x(t), y(t))x′(t) + fy(x(t), y(t))y

′(t) using the single variable chain ruletwice. Here is the proof of the later, when we differentiate f with respect to t and y is treated asa constant:

f( x(t+h) )− f(x(t))

h=

[f( x(t) + (x(t+h)-x(t)) )− f(x(t))]

[x(t+h)-x(t)]·[x(t+h)-x(t)]

h.

Write H(t) = x(t+h)-x(t) in the first part on the right hand side.

f(x(t+ h))− f(x(t))

h=

[f(x(t) +H)− f(x(t))]

H· x(t + h)− x(t)

h.

As h → 0, we also have H → 0 and the first part goes to f ′(x(t)) and the second factor to x′(t).

2 We move on a circle ~r(t) = 〈cos(t), sin(t)〉 on a table with temperature distribution f(x, y) =

x2 − y3. Find the rate of change of the temperature ∇f(x, y) = 〈2x,−3y2〉, ~r′(t) =

〈− sin(t), cos(t)〉 d/dtf(~r(t)) = ∇T (~r(t)) · ~r′(t) = 〈2 cos(t),−3 sin(t)2〉 · 〈− sin(t), cos(t)〉 =−2 cos(t) sin(t)− 3 sin2(t) cos(t).

From f(x, y) = 0 one can express y as a function of x. From d/df(x, y(x)) = ∇f · (1, y′(x)) =fx + fyy

′ = 0, we obtain

Implicit differentation: y′ = −fx/fy.

Even so, we do not know y(x), we can compute its derivative! Implicit differentiation works alsoin three variables. The equation f(x, y, z) = c defines a surface. Near a point where fz is not zero,the surface can be described as a graph z = z(x, y). We can compute the derivative zx withoutactually knowing the function z(x, y). To do so, we consider y a fixed parameter and computeusing the chain rule fx(x, y, z(x, y))1 + fz(x, y)zx(x, y) = 0. This leads to the following

Implicit differentiation:zx(x, y) = −fx(x, y, z)/fz(x, y, z) zy(x, y) = −fy(x, y, z)/fz(x, y, z)

3 The surface f(x, y, z) = x2 + y2/4 + z2/9 = 6 is an ellipsoid. Compute zx(x, y) at the point

(x, y, z) = (2, 1, 1). Solution: zx(x, y) = −fx(2, 1, 1)/fz(2, 1, 1) = −4/(2/9) = −18.

4 How does the chain rule relate to other differentiation rules? Answer. The chain ruleis universal: it implies single variable differentiation rules like the addition, product and

quotient rule in one dimensions:f(x, y) = x+ y, x = u(t), y = v(t), d/dt(x+ y) = fxu

′ + fyv′ = u′ + v′.

f(x, y) = xy, x = u(t), y = v(t), d/dt(xy) = fxu′ + fyv

′ = vu′ + uv′.f(x, y) = x/y, x = u(t), y = v(t), d/dt(x/y) = fxu

′ + fyv′ = u′/y − v′u/v2.

5 Can one prove the chain rule from linearization and just verifying it for linear functions?

Solution. Yes, as in one dimensions, the chain rule follows from linearization. If f is alinear function f(x, y) = ax+ by − c and if the curve ~r(t) = 〈x0 + tu, y0 + tv〉 parametrizes

a line. Then ddtf(~r(t)) = d

dt(a(x0 + tu) + b(y0 + tv)) = au + bv and this is the dot product

of ∇f = (a, b) with ~r ′(t) = (u, v). Since the chain rule only refers to the derivatives of the

functions which agree at the point, the chain rule is also true for general functions.

6 Mechanical systems are determined by the energy function H(x, y), which is a function of

two variables. The first variable x is the position and the second variable y is the momentum.The equations of motion for the curve ~r(t) = 〈x(t), y(t)〉 are called Hamilton equations:

x′(t) = Hy(x, y)

y′(t) = −Hx(x, y)

In a homework you verify that the energy of a Hamiltonian system is preserved: for every

path ~r(t) = 〈x(t), y(t)〉 solving the system, we have H(x(t), y(t)) = const.


16: Gradient and Tangent

The gradient ∇f(x, y) = 〈fx(x, y), fy(x, y)〉 or ∇f(x, y, z) = 〈fx(x, y, z), fy(x, y, z), fz(x, y, z)〉 inthree dimensions is an important object. The symbol ∇ is spelled ”Nabla” and named after anEgyptian harp. The following theorem is important because it provides a crucial link betweencalculus and geometry.

Gradient theorem: Gradients are orthogonal to level curves and level surfaces.

Proof. Every curve ~r(t) on the level curve or level surface satisfies ddtf(~r(t)) = 0. By the chain

rule, ∇f(~r(t)) is perpendicular to the tangent vector ~r′(t). Because this is true for every curve,the gradient is perpendicular to the surface.

The gradient theorem is useful for example because it allows get tangent planes and tangent linesfaster:

The tangent plane through (x0, y0, z0) to a level surface of f(x, y, z) is ax+by+cz = d,where ∇f(x0, y0, z0) = 〈a, b, c〉 and d is obtained by plugging in the point.

The statement in two dimensions is completely analog.

1 Find the tangent plane to the surface 3x2y + z2 − 4 = 0 at the point (1, 1, 1). Solution:

∇f(x, y, z) = 〈6xy, 3x2, 2z〉. And ∇f(1, 1, 1) = 〈6, 3, 2〉. The plane is 6x+3y+2z = d whered is a constant. We can find the constant d by plugging in a point and get 6x+3y+2z = 11.

2 Problem: reflect the ray ~r(t) = 〈1 − t,−t, 1〉 at the surface x4 + y2 + z6 = 6. Solution:

~r(t) hits the surface at the time t = 2 in the point (−1,−2, 1). The velocity vector in that

ray is ~v = 〈−1,−1, 0〉 The normal vector at this point is ∇f(−1,−2, 1) = 〈−4, 4, 6〉 = ~n.The reflected vector is

R(~v = 2Proj~n(~v)− ~v .

We have Proj~n(~v) = 8/68〈−4,−4, 6〉. Therefore, the reflected ray is ~w = (4/17)〈−4,−4, 6〉−〈−1,−1, 0〉.

Lecture 16: Tangent spaces

1 Lets compute the tangent line at (π, 0) to the curve y = sin(x)

directly by determining the slope and making sure the line goes

through the point.

2 Look at f (x, y) = y − sin(x) = 0. Find the gradient ∇f (π, 0) =

〈a, b〉 of f at (π, 0). Now find the tangent line again.

3 Find the tangent plane to the surface x2 − y2 + z2 = −1 at the

point (2, 3, 2).

4 Find a line perpendicular to the surface x2− y2+ z2 = −1 at the

point (2, 3, 2)


17: Extrema

An important problem in multi-variable calculus is to extremize a function f(x, y) of two vari-ables. As in one dimensions, in order to look for maxima or minima, we consider points, wherethe ”derivative” is zero.

A point (a, b) is called a critical point of f(x, y) if ∇f(a, b) = 〈0, 0〉.

Critical points are candidates for extrema because at critical points, all directionalderivatives D~vf = ∇f · ~v are zero. We can not increase the value of f by movinginto any direction.

1

1 Find the critical points of f(x, y) = x4 + y4 − 4xy + 2. The gradient is ∇f(x, y) =〈4(x3 − y), 4(y3 − x)〉 with critical points (0, 0), (1, 1), (−1,−1).

2 f(x, y) = sin(x2 + y) + y. The gradient is ∇f(x, y) = 〈2x cos(x2 + y), cos(x2 + y) + 1〉. For

a critical points, we must have x = 0 and cos(y) + 1 = 0 which means π + k2π. The criticalpoints are at ...(0,−π), (0, π), (0, 3π), ....

3 The graph of f(x, y) = (x2 + y2)e−x2−y2 looks like a volcano. The gradient ∇f = 〈2x −

2x(x2 + y2), 2y − 2y(x2 + y2)〉e−x2−y2 vanishes at (0, 0) and on the circle x2 + y2 = 1. This

function has infinitely many critical points.

4 The function f(x, y) = y2/2−g cos(x) is the energy of the pendulum. The variable g is a con-stant. We have∇f = (y,−g sin(x)) = 〈(0, 0〉 for (x, y) = . . . , (−π, 0), (0, 0), (π, 0), (2π, 0), . . ..

These points are angles for which the pendulum is at rest.

5 The function f(x, y) = |x| + |y| is differentiable on the first quadrant. It does not have

critical points there. The function has a minimum at (0, 0) but it is not in the domain,where the gradient ∇f is defined.

In one dimension, we needed f ′(x) = 0, f ′′(x) > 0 to have a local minimum, f ′(x) = 0, f ′′(x) < 0for a local maximum. If f ′(x) = 0, f ′′(x) = 0, then the critical point was undetermined and couldbe a maximum like for f(x) = −x4, or a minimum like for f(x) = x4 or a flat inflection point likefor f(x) = x3.

Let f(x, y) be a function of two variables with a critical point (a, b). Define D =fxxfyy − f 2

xy. It is called the discriminant of the critical point.

1This definition does not include points, where f or its derivative is not defined. We usually assume that

functions are nice.

Remark: it can be remembered better if knowing that it is the determinant of theHessian matrix

H =

[

fxx fxyfyx fyy

]

.

Second derivative test. Assume (a, b) is a critical point for f(x, y).If D > 0 and fxx(a, b) > 0 then (a, b) is a local minimum.If D > 0 and fxx(a, b) < 0 then (a, b) is a local maximum.If D < 0 then (a, b) is a saddle point.

In the case D = 0, we need higher derivatives to determine the nature of the critical point.

6 The function f(x, y) = x3/3−x− (y3/3−y) has a graph which looks like a ”napkin”. It has

the gradient∇f(x, y) = 〈x2−1,−y2+1〉. There are 4 critical points (1, 1),(−1, 1),(1,−1) and

(−1,−1). The Hessian matrix which includes all partial derivatives is H =

[

2x 0

0 −2y

]

.

For (1, 1) we have D = −4 and so a saddle point,For (−1, 1) we have D = 4, fxx = −2 and so a local maximum,

For (1,−1) we have D = 4, fxx = 2 and so a local minimum.

For (−1,−1) we have D = −4 and so a saddle point. The function has a local maximum, alocal minimum as well as 2 saddle points.

To determine the maximum or minimum of f(x, y) on a domain, determine all critical points inthe interior the domain, and compare their values with maxima or minima at the boundary.We will see next time how to look for extrema on the boundary.

7 Find the maximum of f(x, y) = 2x2 − x3 − y2 on y ≥ −1. With ∇f(x, y) = 4x− 3x2,−2y),

the critical points are (4/3, 0) and (0, 0). The Hessian is H(x, y) =

[

4− 6x 00 −2

]

. At

(0, 0), the discriminant is −8 so that this is a saddle point. At (4/3, 0), the discriminantis 8 and H11 = 4/3, so that (4/3, 0) is a local maximum. We have now also to look at the

boundary y = −1 where the function is g(x) = f(x,−1) = 2x2 − x3 − 1. Since g′(x) = 0 atx = 0, 4/3, where 0 is a local minimum, and 4/3 is a local maximum on the line y = −1.

Comparing f(4/3, 0), f(4/3,−1) shows that (4/3, 0) is the global maximum.


18: Lagrange multipliers

We aim to find maxima and minima of a function f(x, y) in the presence of a constraint g(x, y) =0. A necessary condition for a critical point is that the gradients of f and g are parallel becauseotherwise the we can move along the curve g and increase f . The directional derivative of f in thedirection tangent to the level curve is zero if and only if the tangent vector to g is perpendicularto the gradient of f or if there is no tangent vector.

The system of equations ∇f(x, y) = λ∇g(x, y), g(x, y) = 0 for the three unknownsx, y, λ are called Lagrange equations. The variable λ is a Lagrange multiplier.

Lagrange theorem: Extrema of f(x, y) on the curve g(x, y) = c are either solutionsof the Lagrange equations or critical points of g.

Proof. The condition that ∇f is parallel to ∇g either means ∇f = λ∇g or ∇f = 0 or ∇g = 0.The case ∇f = 0 can be included in the Lagrange equation case with λ = 0.

1 Minimize f(x, y) = x2 + 2y2 under the constraint g(x, y) = x + y2 = 1. Solution: TheLagrange equations are 2x = λ, 4y = λ2y. If y = 0 then x = 1. If y 6= 0 we can divide the

second equation by y and get 2x = λ, 4 = λ2 again showing x = 1. The point x = 1, y = 0is the only solution.

2 Find the shortest distance from the origin to the curve x6 + 3y2 = 1. Solution: Minimizethe function f(x, y) = x2 + y2 under the constraint g(x, y) = x6 + 3y2 = 1. The gradients

are ∇f = 〈2x, 2y〉,∇g = 〈6x5, 6y〉. The Lagrange equations ∇f = λ∇g lead to the system2x = λ6x5, 2y = λ6y, x6 + 3y2 − 1 = 0. We get λ = 1/3, x = x5, so that either x = 0 or 1

or −1. From the constraint equation g = 1, we obtain y =√

(1− x6)/3. So, we have the

solutions (0,±√

1/3) and (1, 0), (−1, 0). To see which is the minimum, just evaluate f on

each of the points. We see that (0,±√

1/3) are the minima.

3 Which cylindrical soda cans of height h and radius r has minimal surface for fixed volume?

Solution: The volume is V (r, h) = hπr2 = 1. The surface area is A(r, h) = 2πrh + 2πr2.With x = hπ, y = r, you need to optimize f(x, y) = 2xy + 2πy2 under the constrained

g(x, y) = xy2 = 1. Calculate ∇f(x, y) = (2y, 2x+ 4πy),∇g(x, y) = (y2, 2xy). The task is tosolve 2y = λy2, 2x+ 4πy = λ2xy, xy2 = 1. The first equation gives yλ = 2. Putting that in

the second one gives 2x+4πy = 4x or 2πy = x. The third equation finally reveals 2πy3 = 1or y = 1/(2π)1/3, x = 2π(2π)1/3. This means h = 0.54.., r = 2h = 1.08.

4 On the curve g(x, y) = x2 − y3 the function f(x, y) = x obviously has a minimum (0, 0).

The Lagrange equations ∇f = λ∇g have no solutions. This is a case where the minimum is

a solution to ∇g(x, y) = 0.

Remarks.

1) Either of the two properties equated in the Lagrange theorem are equivalent to ∇f ×∇g = 0in dimensions 2 or 3.2) With g(x, y) = 0, the Lagrange equations can also be written as ∇F (x, y, λ) = 0 whereF (x, y, λ) = f(x, y)− λg(x, y).3) Either of the two properties equated in the Lagrange theorem are equivalent to ”∇g = λ∇f orf has a critical point”.4) Constrained optimization problems work also in higher dimensions. The proof is the same:

Extrema of f(~x) under the constraint g(~x) = c are either solutions of the Lagrangeequations ∇f = λ∇g, g = c or points where ∇g = ~0.

5 Find the extrema of f(x, y, z) = z on the sphere g(x, y, z) = x2 + y2 + z2 = 1. Solution:

compute the gradients ∇f(x, y, z) = (0, 0, 1), ∇g(x, y, z) = (2x, 2y, 2z) and solve (0, 0, 1) =∇f = λ∇g = (2λx, 2λy, 2λz), x2 + y2 + z2 = 1. The case λ = 0 is excluded by the third

equation 1 = 2λz so that the first two equations 2λx = 0, 2λy = 0 give x = 0, y = 0. The4’th equation gives z = 1 or z = −1. The minimum is the south pole (0, 0,−1) the maximum

the north pole (0, 0, 1).

6 A dice shows k eyes with probability pk. Introduce the vector (p1, p2, p3, p4, p5, p6) with

p1 + p2 + p3 + p4 + p5 + p6 = 1. The entropy of ~p is defined as f(~p) = −∑6

i=1 pi log(pi) =−p1 log(p1)− p2 log(p2)− ...− p6 log(p6). Find the distribution p which maximizes entropy

under the constrained g(~p) = p1 + p2 + p3 + p4 + p5 + p6 = 1. Solution: ∇f = (−1 −log(p1), . . . ,−1 − log(pn)), ∇g = (1, . . . , 1). The Lagrange equations are −1 − log(pi) =

λ, p1+ ...+p6 = 1, from which we get pi = e−(λ+1). The last equation 1 =∑

i exp(−(λ+1)) =6 exp(−(λ+1)) fixes λ = − log(1/6)−1 so that pi = 1/6. The fair dice has maximal entropy.

Maximal entropy means least information content. An unfair dice provides additionalinformation and allows a cheating gambler or casino to gain profit.

7 Assume that the probability that a physical or chemical system is in a state k is pk andthat the energy of the state k is Ek. Nature minimizes the free energy f(p1, . . . , pn) =

−∑

i[pi log(pi)−Eipi] if the energies Ei are fixed. The probability distribution pi satisfying∑

i pi = 1 minimizing the free energy is called a Gibbs distribution. Find this distribution

in general if Ei are given. Solution: ∇f = (−1 − log(p1) − E1, . . . ,−1 − log(pn) − En),∇g = (1, . . . , 1). The Lagrange equation are log(pi) = −1 − λ − Ei, or pi = exp(−Ei)C,

where C = exp(−1− λ). The constraint p1+ · · ·+ pn = 1 gives C(∑

i exp(−Ei)) = 1 so that

C = 1/(∑

i e−Ei). The Gibbs solution is pk = exp(−Ek)/

∑

i exp(−Ei).1

1This example appears in a book of Rufus Bowen, Lecture Notes in Math, 470, 1978


20: Global Extrema

To determine the maximum or minimum of f(x, y) on a domain, we determine all critical points inthe interior the domain, and compare their values with maxima or minima at the boundary.We have to solve both extrema problems with constraints and without constraints.

A point (a, b) is called a global maximum of f(x, y) if f(x, y) ≤ f(a, b) for all(x, y). For example, the point (0, 0) is a global maximum of the function f(x, y) =1− x2 − y2. Similarly, we call (a, b) a global minimum, if f(x, y) ≥ f(a, b) for all(x, y).

1 Does the function f(x, y) = x4+y4−2x2−2y2 have a global maximum or a global minimum?

If yes, find them. Solution: the function has no global maximum. This can be seenby restricting the function to the x-axis, where f(x, 0) = x4 − 2x2 is a function without

maximum. The function has four global minima however. They are located on the 4 points(±1,±1). The best way to see this is to note that f(x, y) = (x2 − 1)2 + (y − 1)2 − 2 which

is minimal when x2 = 1, y2 = 1.

2 Find the maximum of f(x, y) = 2x2 −x3− y2 on y ≥ −1. Solution. With ∇f(x, y) = 4x−3x2,−2y), the critical points are (4/3, 0) and (0, 0). The Hessian isH(x, y) =

[

4− 6x 0

0 −2

]

.

At (0, 0), the discriminant is −8 so that this is a saddle point. At (4/3, 0), the discriminant

is 8 and H11 = 4/3, so that (4/3, 0) is a local maximum. We have now also to look at theboundary y = −1 where the function is g(x) = f(x,−1) = 2x2 − x3 − 1. Since g′(x) = 0 at

x = 0, 4/3, where 0 is a local minimum, and 4/3 is a local maximum on the line y = −1.Comparing f(4/3, 0), f(4/3,−1) shows that (4/3, 0) is the global maximum.

3 Find all extrema of the function f(x, y) = x3 + y3 − 3x − 12y + 20 on the plane andcharacterize them. Do you find a global maximum or global minimum among them? Solu-

tion. The critical points satisfy ∇f(x, y) = 〈0, 0〉 or 〈3x2−3, 3y2−12〉 = 〈0, 0〉. There are 4critical points (x, y) = (±1,±2). The discriminant is D = fxxfyy−f 2

xy = 36xy and fxx = 6x.

point D fxx classification value

(-1,-2) 72 -6 maximum 38

(-1, 2) -72 -6 saddle 6

( 1, -2) -72 6 saddle 34

( 1, 2) 72 6 minimum 2

There are no global maxima nor global minima because the function takes arbitrarily large

and small values. For y = 0 the function is g(x) = f(x, 0) = x3 − 3x + 20 which satisfies

limx→±∞ g(x) = ±∞.

You can ignore the following Q&A safely. But it might answer some of your questions.1. Do global extrema always exist? Yes, if the region Y is compact meaning that for everysequence xn, yn we can pick a subsequence which converges in Y . This is equivalent that thedomain is closed and bounded.

Bolzano’s extremal value theorem. Every continuous function on a compactdomain has a global maximum and a global minimum.

2. Why are critical points important? Critical points are relevant in physics because theyrepresent configurations with lowest energy. Many physical laws describe extrema. The New-ton equations mr′′(t)/2 − ∇V (r(t)) = 0 describing a particle of mass m moving in a field Valong a path γ : t 7→ ~r(t) are equivalent to the property that the path extremizes the are lengthS(γ) =

∫ ba mr′(t)2/2− V (r(t)) dt among all paths.

3. Why is the second derivative test true? Assume f(x, y) has the critical point (0, 0) and

is a quadratic function satisfying f(0, 0) = 0. Then ax2 + 2bxy + cy2 = a(x+ bay)2 + (c− b2

a)y2 =

a(A2 +DB2) with A = (x+ bay), B = b2/a2 and discriminant D. You see that if a = fxx > 0 and

D > 0 then c − b2/a > 0 and the function has positive values for all (x, y) 6= (0, 0). The point(0, 0) is a minimum. If a = fxx < 0 and D > 0, then c− b2/a < 0 and the function has negativevalues for all (x, y) 6= (0, 0) and the point (x, y) is a local maximum. If D < 0, then f takes bothnegative and positive values near (0, 0). For a general function approximate by a quadratic one.

4. Is there something cool about critical points? Yes, assume f(x, y) be the height of anisland. Assume there are only finitely many critical points and all of them have nonzero determi-nant. Label each critical point with a +1 if it is a maximum or minimum, and with −1 if it is asaddle point. The sum of all these numbers is 1, independent of the island. 1

5) Can we avoid Lagrange by substitution? To extremize f(x, y) under the constraint g(x, y) =

0 we find y = y(x) from the second equation and extremize the single variable problem f(x, y(x)).To extremize f(x, y) = y on x2 + y2 = 1 for example we need to extremize

√1− x2. We can

differentiate to get the critical points but also have to look at the cases x = 1 and x = −1, wherethe actual minima and maxima occur. In general also, we can not do the substitution.

6) Is there a second derivative test for Lagrange? A second derivative test can be designed

using second directional derivative in the direction of the tangent. Instead, we just make a list ofcritical points and pick the maximum and minimum.

7) Does Lagrange also work with more constraints? With two constraints the constraint

g = c, h = d defines a curve. The gradient of f must now be in the plane spanned by the gradientsof g and h because otherwise, we could move along the curve and increase f . Here is a formula-tion in three dimensions. Extrema of f(x, y, z) under the constraint g(x, y, z) = c, h(x, y, z) = dare either solutions of the Lagrange equations ∇f = λ∇g + µ∇h, g = c, h = d or solutions to∇g = 0,∇f(x, y, z) = µ∇h, h = d or solutions to ∇h = 0,∇f = λ∇g, g = c or solutions to∇g = ∇h = 0.

8) Why do D and fxx appear in the second derivative test . They are natural. The dis-

criminant D is a determinant det(H) of the matrix H =

[

fxx fxyfyx fyy

]

. If D > 0 then the sign of

fxx is the same as the sign of the trace fxx + fyy which is coordinate independent too. The deter-minant is the product λ1λ2 of the eigenvalues of H and the trace is the sum of the eigenvalues.

9) What does D mean? The discriminant D is defined also at points where we have no critical

point. The number K = D/(1 + |∇f |2)2 is called the Gaussian curvature of the surface. Atcritical points K = D. Curvature is remarkable quantity since it only depends on the intrinsicgeometry of the surface and not on the way how the surface is embedded in space. 2

10) Is there a 2. derivative test in higher dimensions? Yes. one can form the second deriva-

tive matrix H and look at all the eigenvalues of H . If all eigenvalues are negative, we have a localmaximum, if all eigenvalues are positive, we have a local minimum. In general eigenvalues havedifferent signs and we have a saddle point type.

1This follows from the Poincare-Hopf theorem.

2This is the Theorema Egregia of Gauss.


21: Double integrals

The integral∫ ∫

R f(x, y) dxdy is defined as the limit of the Riemann sum

1

n2

∑( in,jn)∈R

f(i

n,j

n)

when n → ∞.

1 If we integrate f(x, y) = xy over the unit square we can sum up the Riemann sum for fixed

y = j/n and get y/2. Now perform the integral over y to get 1/4. This example shows howwe can reduce double integrals to single variable integrals.

2 If f(x, y) = 1, then the integral is the area of the region R. The integral is the limit L(n)/n2,where L(n) is the number of lattice points (i/n, j/n) inside R.

3 The integral∫ ∫

R f(x, y) dxdy as the signed volume of the solid below the graph of f

and above the region R in the x − y plane. The volume below the xy-plane is countednegatively.

Fubini’s theorem allows to switch the order of integration over a rectangle, if thefunction f is continuous:

∫ ba

∫ dc f(x, y) dxdy =

∫ dc

∫ ba f(x, y) dydx.

Proof. For every n the ”quantum Fubini identity”

∑in∈[a,b]

∑jn∈[c,d]

f(i

n,j

n) =

∑jn∈[c,d]

∑in∈[a,b]

f(i

n,j

n)

holds for all functions. Now divide both sides by n2 and take the limit n → ∞.

A type I region is of the form

R = {(x, y) | a ≤ x ≤ b, c(x) ≤ y ≤ d(x) } .

An integral over such a region is called a type I integral

∫∫Rf dA =

∫ b

a

∫ d(x)

c(x)f(x, y) dydx .

ba

cHxL

d HxL

A type II region is of the form

R = {(x, y) | c ≤ y ≤ d, a(y) ≤ x ≤ b(y) } .

An integral over such a region is called a type II inte-

gral ∫∫Rf dA =

∫ d

c

∫ b(y)

a(y)f(x, y) dxdy .

d

c

a HyL

b HyL

4 Integrate f(x, y) = x2 over the region bounded aboveby sin(x3) and bounded below by the graph of − sin(x3)

for 0 ≤ x ≤ π. The value of this integral has a physicalmeaning. It is called moment of inertia.

∫ π1/3

0

∫ sin(x3)

− sin(x3)x2 dydx = 2

∫ π1/3

0sin(x3)x2 dx

This can be solved by substitution

= −2

3cos(x3)|π

1/3

0 =4

3.

5 Integrate f(x, y) = y2 over the region bound by the x-axes, the lines y = x+ 1 and y = 1− x. The problem is

best solved as a type I integral. because we would haveto compute 2 different integrals as a type I integral. The

y bounds are x = y − 1 and x = 1− y

∫ 1

0

∫ 1−y

y−1y3 dx dy = 2

∫ 1

0y3(1−y) dy = 2(

1

4−

1

3) =

1

10.

6 Let R be the triangle 1 ≥ x ≥ 0, 0 ≤ y ≤ x. What is

∫ ∫Re−x2

dxdy ?

The type II integral∫ 10 [∫ 1y e−x2

dx]dy can not be solved

because e−x2

has no anti-derivative in terms of elemen-tary functions. The type I integral

∫ 10 [∫ x0 e−x2

dy] dx

however can be solved:

=∫ 1

0xe−x2

dx = −e−x2

2|10 =

(1− e−1)

2= 0.316... .


Lecture 21: Polar integration

1 The area of a disc of radius R is

∫ R

−R

∫

√

R2−x2

−

√

R2−x2

1 dydx =∫ R

−R2√R2 − x2 dx .

This integral can be solved with the substitution x =

R sin(u), dx = R cos(u)

∫ π/2

−π/22√

R2 −R2 sin2(u)R cos(u) du =∫ π/2

−π/22R2 cos2(u) du .

Using a double angle formula we get

R2∫ π/2−π/2 2

(1+cos(2u)2

du = R2π. We will now seehow to do that better in polar coordinates.

A polar region is a region bound by a simple closed curve given in polar coordinatesas the curve (r(t), θ(t)).

In Cartesian coordinates the parametrization of the boundary curve is ~r(t) = 〈r(t) cos(θ(t), r(t) sin(θ(t)〉.We are especially interested in regions which are bound by polar graphs, where θ(t) = t.

2 The polar region defined by r ≤ | cos(3θ)| belongs to the class of roses r(t) = | cos(nt)|they are also called rhododenea. These names reflect that polar regions model flowerswell.

3 The polar curve r(θ) = 1 + sin(θ) is called a cardioid. It looks like a heart. It is a special

case of a limacon a polar curve of the form r(θ) = 1 + b sin(θ).

4 The polar curve r(θ) = |√

cos(2t)| is called a lemniscate. It looks like an infinity sign. It

encloses a flower with two petals.

-1.0 -0.5 0.5 1.0

0.5

1.0

1.5

2.0

-1.0 -0.5 0.5 1.0

-0.5

0.5

-1.0 -0.5 0.5 1.0

-1.0

-0.5

0.5

1.0

To integrate in polar coordinates, we evaluate the integral

∫∫

Rf(x, y) dxdy =

∫∫

Rf(r cos(θ), r sin(θ)r drdθ

5 Integratef(x, y) = x2 + x2 + xy ,

over the unit disc. We have f(x, y) = f(r cos(θ), r sin(θ)) = r2 + r2 cos(θ) sin(θ) so that∫∫

Rf(x, y) dxdy =∫ 10

∫ 2π0 (r2 + r2 cos(θ) sin(θ))r dθdr = 2π/4.

6 We have earlier computed area of the disc {x2 + y2 ≤ R2 } using substitution. It is moreelegant to do this integral in polar coordinates:

∫ 2π0

∫ R0 r drdθ = 2πr2/2|R0 = πR2.

Why do we have to include the factor r, when we move to polar coordinates? The reason isthat a small rectangle R with dimensions dθdr in the (r, θ) plane is mapped by T : (r, θ) 7→(r cos(θ), r sin(θ)) to a sector segment S in the (x, y) plane. It has the area r dθdr.

7 Integrate the function f(x, y) = 1 {(θ, r(θ)) | r(θ) ≤ | cos(3θ)| }.∫ ∫

R1 dxdy =

∫ 2π

0

∫ cos(3θ)

0r dr dθ =

∫ 2π

0

cos(3θ)2

2dθ = π/2 .

8 Integrate f(x, y) = y√x2 + y2 over the region R = {(x, y) | 1 < x2 + y2 < 4, y > 0 }.

∫ 2

1

∫ π

0r sin(θ)r r dθdr =

∫ 2

1r3

∫ π

0sin(θ) dθdr =

(24 − 14)

4

∫ π

0sin(θ) dθ = 15/2

For integration problems, where the region is part of an annular region, or if you see function

with terms x2 + y2 try to use polar coordinates x = r cos(θ), y = r sin(θ).

9 The Belgian Biologist Johan Gielis defined in 1997 with the family of curves given in polarcoordinates as

r(φ) = (| cos(mφ

4)|n1

a+

| sin(mφ4)|n2

b)−1/n3

This super-curve can produce a variety of shapes like circles, square, triangle, stars. It

can also be used to produce ”super-shapes”. The super-curve generalizes the super-ellipsewhich had been discussed in 1818 by Lame and helps to describe forms in biology. 1

1”Gielis, J. A ’generic geometric transformation that unifies a wide range of natural and abstract shapes’.American Journal of Botany, 90, 333 - 338, (2003).


Lecture 22: Surface area

A surface ~r(u, v) parametrized on a parameter domain R has the surface area

∫ ∫

R|~ru(u, v)× ~rv(u, v)| dudv .

Proof. The vector ~ru is tangent to the grid curve u 7→ ~r(u, v) and ~rv is tangent to v 7→ ~r(u, v).The two vectors span a parallelogram with area |~ru×~rv|. A small rectangle [u, u+du]× [v, v+dv]is mapped by ~r to a parallelogram spanned by [~r, ~r + ~ru] and [~r, ~r + ~rv] which has the area|~ru(u, v)× ~rv(u, v)| dudv.

1 The parametrized surface ~r(u, v) = 〈2u, 3v, 0〉 is part of the xy-plane. The parameter regionG just gets stretched by a factor 2 in the x coordinate and by a factor 3 in the y coordinate.

~ru × ~rv = 〈0, 0, 6〉 and we see for example that the area of ~r(G) is 6 times the area of G.

For a planar region ~r(s, t) = P + sv + tw where (s, t) ∈ G, the surface area is thearea of G times |v × w|.

2 The map ~r(u, v) = 〈L cos(u) sin(v), L sin(u) sin(v), L cos(v)〉maps the rectangle G = [0, 2π]×[0, π] onto the sphere of radius L. We compute ~ru × ~rv = L sin(v)~r(u, v). So, |~ru × ~rv| =L2| sin(v)| and ∫ ∫

R 1 dS =∫

2π0

∫ π0L2 sin(v) dvdu = 4πL2

For a sphere of radius L, we have |~ru × ~rv| = L2 sin(v) The surface area is 4πL2.

3 For graphs (u, v) 7→ 〈u, v, f(u, v)〉, we have ~ru = (1, 0, fu(u, v)) and ~rv = (0, 1, fv(u, v)). The

cross product ~ru × ~rv = (−fu,−fv, 1) has the length√

1 + f 2u + f 2

v . The area of the surface

above a region G is∫ ∫

G

√

1 + f 2u + f 2

v dudv.

For a graph z = f(x, y) parametrized over G, the surface area is

∫ ∫

G

√

1 + f 2x + f 2

y dxdy .

4 Lets take a surface of revolution ~r(u, v) = 〈v, f(v) cos(u), f(v) sin(u)〉 on R = [0, 2π] ×[a, b]. We have ~ru = (0,−f(v) sin(u), f(v) cos(u)), ~rv = (1, f ′(v) cos(u), f ′(v) sin(u)) and

~ru × ~rv = (−f(v)f ′(v), f(v) cos(u), f(v) sin(u)) = f(v)(−f ′(v), cos(u), sin(u)). The surface

area is∫ ∫ |~ru × ~rv| dudv = 2π

∫ ba |f(v)|

√

1 + f ′(v)2 dv.

For a surface of revolution r = f(z) with a ≤ z ≤ b, the surface area is

2π∫ b

a|f(z)|

√

1 + f ′(z)2 dz .

5

Gabriel’s trumpet is the surface of revolution where

g(z) = 1/z, where 1 ≤ z < ∞. Its volume is∫

∞

1πg(z)2 dz = π. We will compute in class the sur-

face area.

6 Find the surface area of the part of the paraboloid x = y2 + z2 which is inside the cylindery2 + z2 ≤ 9. Solution. We use polar coordinates in the yz-plane. The paraboloid is

parametrized by (u, v) 7→ (v2, v cos(u), v sin(u)) and the surface integral∫

3

0

∫

2π0

|~ru×~rv| dudvis equal to

∫

3

0

∫

2π0

v√1 + 4v2 dudv = 2π

∫

3

0v√1 + 4v2 dv = π(373/2 − 1)/6.

7 In this example we derive the distortion factor r in polar coordinates. To do so, weparametrize a region in the xy plane with ~r(u, v) = 〈u cos(v), u sin(v), 0〉. Given a region

G in the uv plane like the rectangle [0, π] × [1, 2], we obtain a region S in the xy plane asthe image. The factor |~ru × ~rv| is equal to the radius u. In our example, the surface area is∫ π0

∫

2

1u dudv = π(4 − 1) = 3π. This is the area of the half annulus S. We could have used

polar coordinates directly in the xy plane and compute∫ π0

∫

2

1r dr dθ = 3π. But the only

thing which has changed are the names of the variables.

8

The surface parametrized by

~r(u, v) = 〈(2+v cos(u/2)) cos(u), (2+v cos(u/2)) sin(u), v sin(u/2)〉

on G = [0, 2π] × [−1, 1] is called a Mobius strip.What is its surface area? Solution. The calculation

of |~ru × ~rv|2 = 4 + 3v2/4 + 4v cos(u/2) + v2 cos(u)/2is straightforward but a bit tedious. The integral over

[0, 2π] × [−1, 1] can only be evaluated numerically, the

result is 25.413....


Lecture 24: Triple integrals

If f(x, y, z) is a function of three variables and E is a solid region in space, then∫ ∫ ∫

E f(x, y, z) dxdydz is defined as the n → ∞ limit of the Riemann sum

1

n3

∑

(i/n,j/n,k/n)∈E

f(i

n,j

n,k

n) .

As in two dimensions, triple integrals can be evaluated by iterated 1D integral computations. Hereis a simple example:

1 Assume E is the box [0, 1]× [0, 1]× [0, 1] and f(x, y, z) = 24x2y3z.

∫ 10

∫ 10

∫ 10 24x2y3z dz dy dx .

To compute the integral we start from the core∫ 10 24x2y3z dz = 12x3y3, then integrate the

middle layer,∫ 10 12x3y3 dy = 3x2 and finally and finally handle the outer layer:

∫ 10 3x2dx =1.

When we calculate the most inner integral, we fix x and y. The integral is integrating upf(x, y, z) along a line intersected with the body. After completing the middle integral, we

have computed the integral on the plane z = const intersected with R. The most outer

integral sums up all these two dimensional sections.

The two important methods for triple integrals are the ”washer method” and the ”sandwichmethod”. The washer method from single variable calculus reduces the problem directly

to a one dimensional integral. The new sandwich method reduces the problem to a twodimensional integration problem.

The washer method slices the solid along the z-axes. If g(z) is the double in-tegral along the two dimensional slice, then

∫ ba [∫ ∫

R(z) f(x, y, z) dxdy] dz. The

sandwich method sees the solid sandwiched between the graphs of two functionsg(x, y) and h(x, y) over a common two dimensional region R. The integral becomes∫ ∫

R[∫ h(x,y)g(x,y) f(x, y, z) dz] dxdy.

2 An important special case of the sandwich method is the volume

∫

R

∫ f(x,y)

01 dzdxdy .

under the graph of a function f(x, y) and above a region R. It is the integral∫ ∫

R f(x, y) dA.

What we actually have computed is a triple integral

3 Find the volume of the unit sphere. Solution: The sphere is sandwiched between the graphs

of two functions. Let R be the unit disc in the xy plane. If we use the sandwich method,we get

V =∫ ∫

R[∫

√1−x2−y2

−

√1−x2−y2

1dz]dA .

which gives a double integral∫ ∫

R 2√1− x2 − y2 dA which is of course best solved in polar

coordinates. We have∫ 2π0

∫ 10

√1− r2r drdθ = 4π/3.

With the washer method which is in this case also called disc method, we slice along

the z axes and get a disc of radius√1− z2 with area π(1− z2). This is a method suitable

for single variable calculus because we get directly∫ 1−1 π(1− z2) dz = 4π/3.

4 The mass of a body with density ρ(x, y, z) is defined as∫ ∫ ∫

R ρ(x, y, z) dV . For bodies withconstant density ρ the mass is ρV , where V is the volume. Compute the mass of a body which

is bounded by the parabolic cylinder z = 4− x2, and the planes x = 0, y = 0, y = 6, z = 0 if

the density of the body is 1. Solution:∫ 2

0

∫ 6

0

∫ 4−x2

0dz dy dx =

∫ 2

0

∫ 6

0(4− x2) dydx

= 6∫ 2

0(4− x2) dx = 6(4x− x3/3)|20 = 32

5

The solid region bound by x2 + y2 = 1, x = z and z = 0

is called the hoof of Archimedes. It is historicallysignificant because it is one of the first examples, on

which Archimedes probed his Riemann sum integrationtechnique. It appears in every calculus text book. Find

the volume. Solution. Look from the situation from

above and picture it in the x− y plane. You see a halfdisc R. It is the floor of the solid. The roof is the

function z = x. We have to integrate∫ ∫

R x dxdy. Wegot a double integral problems which is best done in

polar coordinates;∫ π/2−π/2

∫ 10 r2 cos(θ) drdθ = 2/3.

6

Finding the volume of the solid region bound by the

three cylinders x2 + y2 = 1, x2 + z2 = 1 and y2 + z2 = 1is one of the most famous volume integration problems.

Solution: look at 1/16’th of the body given in cylindri-cal coordinates 0 ≤ θ ≤ π/4, r ≤ 1, z > 0. The roof is

z =√1− x2 because above the ”one eighth disc” R only

the cylinder x2 + z2 = 1 matters. The polar integration

problem

16∫ π/4

0

∫ 1

0

√

1− r2 cos2(θ)r drdθ

has an inner r-integral of (16/3)(1 − sin(θ)3)/ cos2(θ).

Integrating this over θ can be done by integrating (1 +sin(x)3) sec2(x) by parts using tan′(x) = sec2(x) leading

to the anti derivative cos(x)+sec(x)+tan(x). The resultis 16− 8

√2.


Lecture 25: Spherical integration

Cylindrical coordinates are coordinates in space in which polar coordinates are chosen in the xyplane and the z-coordinate is left untouched. A surface of revolution can be described in cylindricalcoordinates as r = g(z). The coordinate change transformation T (r, θ, z) = (r cos(θ), r sin(θ), z),produces the same integration factor r as in polar coordinates.

∫∫T (R)

f(x, y, z) dxdydz =∫∫

Rg(r, θ, z) r drdθdz

Remember also that spherical coordinates use ρ, the distance to the origin as well as two angles:θ the polar angle and φ, the angle between the vector and the z axis. The coordinate change is

T : (x, y, z) = (ρ cos(θ) sin(φ), ρ sin(θ) sin(φ), ρ cos(φ)) .

The integration factor can be seen by measuring the volume of a spherical wedge which isdρ, ρ sin(φ) dθ, ρdφ = ρ2 sin(φ)dθdφdρ.

∫∫T (R)

f(x, y, z) dxdydz =∫∫

Rg(ρ, θ, z) ρ2 sin(φ) dρdθdφ

1 A sphere of radius R has the volume

∫ R

0

∫ 2π

0

∫ π

0ρ2 sin(φ) dφdθdρ .

The most inner integral∫ π0 ρ2 sin(φ)dφ = −ρ2 cos(φ)|π0 = 2ρ2. The next layer is, because φ

does not appear:∫ 2π0 2ρ2 dφ = 4πρ2. The final integral is

∫R0 4πρ2 dρ = 4πR3/3.

The moment of inertia of a body G with respect to an axis L is defined as thetriple integral

∫ ∫ ∫G r(x, y, z)2 dzdydx, where r(x, y, z) = R sin(φ) is the distance

from the axis L.

2

For a sphere of radius R we obtain with respect to thez-axis:

I =∫ R

0

∫ 2π

0

∫ π

0ρ2 sin2(φ)ρ2 sin(φ) dφdθdρ

= (∫ π

0sin3(φ) dφ)(

∫ R

0ρ4 dr)(

∫ 2π

0dθ)

= (∫ π

0sin(φ)(1− cos2(φ)) dφ)(

∫ R

0ρ4 dr)(

∫ 2pi

0dθ)

= (− cos(φ)+cos(φ)3/3)|π0(L5/5)(2π) =4

3·R

5

5·2π =

8πR5

15.

If the sphere rotates with angular velocity ω, then Iω2/2 is the kinetic energy of that sphere.Example: the moment of inertia of the earth is 8·1037kgm2. The angular velocity is ω = 2π/day =2π/(86400s). The rotational energy is 8 · 1037kgm2/(7464960000s2) ∼ 1029J ∼ 2.51024kcal.

3 Find the volume and the center of mass of a diamond, the intersection of the unit spherewith the cone given in cylindrical coordinates as z =

√3r.

Solution: we use spherical coordinates to find the center of mass

x =∫ 1

0

∫ 2π

0

∫ π/6

0ρ3 sin2(φ) cos(θ) dφdθdρ

1

V= 0

y =∫ 1

0

∫ 2π

0

∫ π/6

0ρ3 sin2(φ) sin(θ) dφdθdρ

1

V= 0

z =∫ 1

0

∫ 2π

0

∫ π/6

0ρ3 cos(φ) sin(φ) dφdθdρ

1

V=

2π

32V

4

Find∫ ∫ ∫

R z2 dV for the solid obtained by intersecting

{1 ≤ x2 + y2 + z2 ≤ 4 } with the double cone {z2 ≥x2 + y2}.Solution: since the result for the double cone is twicethe result for the single cone, we work with the diamond

shaped region R in {z > 0} and multiply the result atthe end with 2. In spherical coordinates, the solid R is

given by 1 ≤ ρ ≤ 2 and 0 ≤ φ ≤ π/4. With z = ρ cos(φ),we have

∫ 2

1

∫ 2π

0

∫ π/4

0ρ4 cos2(φ) sin(φ) dφdθdρ

= (25

5− 15

5)2π(

− cos3(φ))

3|π/40 = 2π

31

5(1− 2−3/2) .

The result for the double cone is 4π(31/5)(1− 1/√23) .


Lecture 26: Vector fields

A vector field in the plane is a map, which assigns to each point (x, y) a vector~F (x, y) = 〈P (x, y), Q(x, y)〉. A vector field in space is a map, which assigns to

(x, y, z) in space a vector ~F (x, y, z) = 〈P (x, y, z), Q(x, y, z), R(x, y, z)〉.

For example ~F (x, y) = 〈x−1, y〉/((x−1)2+y2)3/2−〈x+1, y〉/((x+1)2+y2)3/2 is the electric fieldof positive and negative point charge. It is called dipole field. It is shown in the picture below:

If f(x, y) is a function of two variables, then ~F (x, y) = ∇f(x, y) is called a gradient

field. Gradient fields in space are of the form ~F (x, y, z) = ∇f(x, y, z).

When is a vector field a gradient field? ~F (x, y) = 〈P (x, y), Q(x, y)〉 = ∇f(x, y) implies Qx(x, y) =Py(x, y). If this does not hold at some point, F is no gradient field.

Clairaut test: if Qx(x, y) − Py(x, y) is not zero at some point, then ~F (x, y) =〈P (x, y), Q(x, y)〉 is not a gradient field.

We will see next week that the condition curl(F ) = Qx − Py = 0 is also necessary for F to bea gradient field. In class, we see more examples on how to construct the potential f from thegradient field F .

1 Is the vector field ~F (x, y) = 〈P,Q〉 = 〈3x2y + y + 2, x3 + x− 1〉 a gradient field? Solution:

the Clairot test shows Qx − Py = 0. We integrate the equation fx = P = 3x2y + y + 2 andget f(x, y) = 2x+ xy + x3y + c(y). Now take the derivative of this with respect to y to get

x + x2 + c′(y) and compare with x3 + x− 1. We see c′(y) = −1 and so c(y) = −y + c. We

see the solution x3y + xy − y + 2x .

2 Is the vector field ~F (x, y) = 〈xy, 2xy2〉 a gradient field? Solution: No: Qx − Py = 2y2 − x

is not zero.

Vector fields are important in differential equations. Motivation comes also from mechanics:

3 A class of vector fields important in mechanics areHamiltonian fields: If H(x, y) is a func-

tion of two variables, then 〈Hy(x, y),−Hx(x, y)〉 is called a Hamiltonian vector field. Anexample is the harmonic oscillator H(x, y) = x2+y2. Its vector field (Hy(x, y),−Hx(x, y)) =

(y,−x). The flow lines of a Hamiltonian vector fields are located on the level curves of H

(as you have shown in th homework with the chain rule).

4 Newton’s law m~r′′ = F relates the acceleration ~r′′ of a body with the force F acting at

the point. For example, if x(t) is the position of a mass point in [−1, 1] attached at two

springs and the mass is m = 2, then the point experiences a force (−x + (−x)) = −2x sothat mx′′ = 2x or x′′(t) = −x(t). If we introduce y(t) = x′(t) of t, then x′(t) = y(t) and

y′(t) = −x(t). Of course y is the velocity of the mass point, so a pair (x, y), thought of asan initial condition, describes the system so that nature knows what the future evolution of

the system has to be given that data.

5 Is the vector field ~F (x, y) = 〈P (x, y), Q(x, y)〉 = 〈xy, x2〉 a gradient field?

No. Is the vector field ~F (x, y) = 〈P (x, y), Q(x, y)〉 = 〈sin(x) + y, cos(y) + x〉 a gradientfield? Yes. the function is f(x, y) = − cos(x) + sin(y) + xy.

6 Can you spot the following vector fields in the pictures? F (x, y) = 〈y, 0〉, F (x, y) = 〈−y −

x, x+ y〉, F (x, y) = 〈−y, x〉, F (x, y) = 〈y − x, x+ y〉. Which ones are conservative?


Lecture 28: Fundamental theorem of line integrals

Recall:

If ~F is a vector field in the plane or in space and C : t 7→ ~r(t) is a curve defined onthe interval [a, b] then ∫ b

a

~F (~r(t)) · ~r ′(t) dt

is called the line integral of ~F along the curve C.

The following theorem generalizes the fundamental theorem of calculus to higher dimensions:

Fundamental theorem of line integrals: If ~F = ∇f , then

∫ b

a

~F (~r(t)) · ~r′(t) dt = f(~r(b))− f(~r(a)) .

The proof of the fundamental theorem uses the chain rule in the second equality and the funda-mental theorem of calculus in the third equality of the following identities:

∫ b

a

~F (~r(t)) · ~r′(t) dt =∫ b

a∇f(~r(t)) · ~r′(t) dt =

∫ b

a

d

dtf(~r(t)) dt = f(~r(b))− f(~r(a)) .

1 Let ~F (x, y) = 〈2xy2 + 3x2, 2yx2〉. Find a potential f of ~F = 〈P,Q〉.

Solution: The potential function f(x, y) satisfies fx(x, y) = 2xy2 + 3x2 and fy(x, y) = 2yx2.Integrating the second equation gives f(x, y) = x2y2 + h(x). Partial differentiation with re-

spect to x gives fx(x, y) = 2xy2+h′(x) which should be 2xy2+3x2 so that we can take h(x) =x3. The potential function is f(x, y) = x2y2+x3. Find g, h from f(x, y) =

∫ x0P (t, y) dt+h(y)

and fy(x, y) = g(x, y).

2 At Oliver’s last birthday, he relaxed in a Jacuzzi of the Boston Harbor hotel. He movedalong curve C which is given by part of the curve x10+y10 = 1 in the first quadrant, oriented

counter clockwise. The hot water in the tub has the velocity ~F (x, y) = 〈x, y4〉. Calculatethe line integral

∫C~F · ~dr, the energy you gain from the fluid force.


Lecture 29: Green’s theorem

The curl of a vector field ~F (x, y) = 〈P (x, y), Q(x, y)〉 is the scalar field

curl(F )(x, y) = Qx(x, y)− Py(x, y) .

The curl(F ) measures the vorticity of the vector field. One can write ∇× ~F = curl(~F ) because

the two dimensional cross product of (∂x, ∂y) with ~F = 〈P,Q〉 is the scalar Qx − Py.

1 For ~F (x, y) = 〈−y, x〉 we have curl(F )(x, y) = 2.

2 If ~F (x, y) = ∇f is a gradient field then the curl is zero because if P (x, y) = fx(x, y), Q(x, y) =

fy(x, y) and curl(F ) = Qx − Py = fyx − fxy = 0 by Clairaut’s theorem.

Green’s theorem: If ~F (x, y) = 〈P (x, y), Q(x, y)〉 is a vector field and R is a regionfor which the boundary C is parametrized so that R is ”to the left”, then

∫C

~F · ~dr =∫ ∫

Gcurl(F ) dxdy .

Proof. The integral of ~F along the boundary of G = [x, x+ǫ]×[y, y+ǫ] is∫ ǫ0P (x+t, y)dt+

∫ ǫ0Q(x+

ǫ, y+ t) dt−∫ ǫ0P (x+ t, y+ ǫ) dt−

∫ ǫ0Q(x, y+ t) dt. Because Q(x+ ǫ, y)−Q(x, y) ∼ Qx(x, y)ǫ and

P (x, y+ ǫ)−P (x, y) ∼ Py(x, y)ǫ, this is is (Qx −Py)ǫ2 ∼

∫ ǫ0

∫ ǫ0curl(F ) dxdy. All identities hold in

the limit ǫ → 0.

A general region G can be cut into small squares ofsize ǫ. Summing up all the line integrals around theboundaries gives the line integral around the boundarybecause in the interior, the line integrals cancel. Sum-ming up the vorticities on the squares is a Riemann sumapproximation of the double integral. The boundaryintegrals converge to the line integral of C.

George Green lived from 1793 to 1841. He was a physicist a self-taught mathematician andmiller.

3 If ~F is a gradient field then both sides of Green’s theorem are zero:∫C~F · ~dr is zero by

the fundamental theorem for line integrals. and∫ ∫

G curl(F ) · dA is zero because curl(F ) =

curl(grad(f)) = 0.

The already established Clairaut identity

curl(grad(f)) = 0

can also remembered as ∇ × ∇f noting that the the cross product of two identical vectors is 0.Treating ∇ as a vector is nabla calculus.

4 Find the line integral of ~F (x, y) = 〈x2−y2, 2xy〉 = 〈P,Q〉 along the boundary of the rectangle

[0, 2]×[0, 1]. Solution: curl(~F ) = Qx−Py = 2y−2y = −4y so that∫C~F ~dr =

∫2

0

∫1

04y dydx =

2y2|10x|2

0= 4.

5

Find the area of the region enclosed by

~r(t) = 〈sin(πt)2

t, t2 − 1〉

for −1 ≤ t ≤ 1. To do so, use Greens theorem with thevector field ~F = 〈0, x〉.

6 An important application of Green is to compute area. With the vector fields ~F (x, y) =

〈P,Q〉 = 〈−y, 0〉 or ~F (x, y) = 〈0, x〉 have vorticity curl(~F )(x, y) = 1. For ~F (x, y) = 〈0, x〉 ,the right hand side in Green’s theorem is the area of G:

Area(G) =∫C〈0, x(t)〉 · 〈x′(t), y′(t)〉 dt .

7 Let G be the region under the graph of a function f(x) on [a, b]. The line integral aroundthe boundary of G is 0 from (a, 0) to (b, 0) because ~F (x, y) = 〈0, 0〉 there. The line integral

is also zero from (b, 0) to (b, f(b)) and (a, f(a)) to (a, 0) because N = 0. The line integral

along the curve (t, f(t)) is −∫ ba 〈−y(t), 0〉·〈1, f ′(t)〉 dt =

∫ ba f(t) dt. Green’s theorem confirms

that this is the area of the region below the graph.

It had been a consequence of the fundamental theorem of line integrals that

If ~F is a gradient field then curl(F ) = 0 everywhere.

Is the converse true? Here is the answer:

A region R is called simply connected if every closed loop in R can be pulledtogether to a point in R.

If curl(~F ) = 0 in a simply connected region G, then ~F is a gradient field.

Proof. Given a closed curve C inG enclosing a regionR. Green’s theorem assures that∫C~F ~dr = 0.

So ~F has the closed loop property in G, line integrals are path independent and ~F is a gradientfield.


Lecture 30: Divergence and Curl

The curl of a vector field ~F = 〈P,Q,R〉 is the vector field

curl(P,Q,R) = 〈Ry −Qz, Pz − Rx, Qx − Py〉 .

1 The curl of the vector field 〈x2 + y5, z2, x2 + z2〉 is 〈−2z,−2x,−5y4〉.

We can write curl(~F ) = ∇× ~F . In two dimensions we had

The curl of a vector field ~F = 〈P,Q〉 is Qx − Py, a scalar field.

If a field has zero curl everywhere, the field is called irrotational.

The curl is often visualized using a ”paddle wheel”. Ifyou place such a wheel into the field into the direction v,its rotation speed of the wheel measures the quantity ~F ·~v. We will see why soon. Consequently, the direction inwhich the wheel turns fastest, is the direction of curl(~F ).Its angular velocity is the length of the curl. The wheelcould actually be used to measure the curl of the vectorfield at any point. In situations with large vorticity likein a tornado, one can ”see” the direction of the curl nearthe vortex center.

In two dimensions, we had two derivatives, the gradient and curl. In three dimensions, there arethree fundamental derivatives, the gradient, the curl and the divergence.

The divergence of ~F = 〈P,Q,R〉 is the scalar field div(〈P,Q,R〉) = ∇ · ~F =Px +Qy +Rz.

The divergence can also be defined in two dimensions, but it is not fundamental.

The divergence of ~F = 〈P,Q〉 is div(P,Q) = ∇ · ~F = Px +Qy.

In two dimensions, the divergence is just the curl of a −90 degrees rotated field ~G = 〈Q,−P 〉

because div( ~G) = Qx − Py = curl(~F ). The divergence measures the ”expansion” of a field. If afield has zero divergence everywhere, the field is called incompressible.

With the ”vector” ∇ = 〈∂x, ∂y, ∂z〉, we can write curl(~F ) = ∇× ~F and div(~F ) = ∇ · ~F .

∆f = div(grad(f)) = fxx + fyy + fzz .

is the Laplacian of f . One also writes ∆f = ∇2f because ∇ · (∇f) = div(grad(f).

From ∇ · ∇ × ~F = 0 and ∇×∇~F = ~0, we get

div(curl(~F )) = 0, curlgrad(~F ) = ~0.

2 Question: Is there a vector field ~G such that ~F = 〈x+ y, z, y2〉 = curl( ~G)?

Answer: No, because div(~F ) = 1 is incompatible with div(curl( ~G)) = 0.

3 Show that in simply connected region, every irrotational and incompressible field can bewritten as a vector field ~F = grad(f) with ∆f = 0. Proof. Since ~F is irrotational, there exists

a function f satisfying F = grad(f). Now, div(F ) = 0 implies divgrad(f) = ∆f = 0.

4 Find an example of a field which is both incompressible and irrotational. Solution. Findf which satisfies the Laplace equation ∆f = 0, like f(x, y) = x3 − 3xy2, then look at its

gradient field ~F = ∇f . In that case, this gives

~F (x, y) = 〈3x2 − 3y2,−6xy〉 .

5 If we rotate the vector field ~F = 〈P,Q〉 by 90 degrees = π/2, we get a new vector field~G = 〈−Q,P 〉. The integral

∫C F · ds becomes a flux

∫γ G · dn of G through the boundary

of R, where dn is a normal vector with length |r′|dt. With div(~F ) = (Px +Qy), we see that

curl(~F ) = div( ~G) and Green’s theorem is∫ ∫

R div( ~G) dxdy =∫C~G · ~dn. where dn(x, y) is a

normal vector at (x, y) orthogonal to the velocity vector ~r ′(x, y) at (x, y). Do not bother

with this, it is just Green’s theorem in disguise.

The divergence at a point (x, y) is the flux of the field through a small circle of

radius r around the point in the limit when the radius of the circle goes to zero

The curl at a point (x, y) is the work done along a small circle of radius r aroundthe point in the limit when the radius of the circle goes to zero

We have now all the derivatives together. In dimension d, there are d fundamental derivatives.

1

1grad−→ 1

1grad−→ 2

curl−→ 1

1grad−→ 3

curl−→ 3

div−→ 1


Lecture 31: Flux integrals

If a surface S is parametrized as ~r(u, v) = 〈x(u, v), y(u, v), z(u, v)〉 over a domain G

in the uv-plane and ~F is a vector field, then the flux integral of ~F through S is

∫ ∫G

~F (~r(u, v)) · (~ru × ~rv) dudv .

With the short hand notation ~dS = (~ru × ~rv) dudv representing an infinitesimal normal vector to

the surface, this can be written as∫ ∫

S~F · ~dS. The interpretation is that if ~F = fluid velocity field,

then∫ ∫

S~F · ~dS is the amount of fluid passing through S in unit time.

Because ~n = ~ru × ~rv/|~ru × ~rv| is a unit vector normal to the surface and on the surface, ~F · ~n isthe normal component of the vector field with respect to the surface. One could write thereforealso

∫ ∫S~F · ~dS =

∫ ∫ ~F · ~n dS where dS is the surface element we know from when we computed

surface area. The function ~F ·~n is the scalar projection of ~F in the normal direction. Whereas theformula

∫ ∫1 dS gave the area of the surface with dS = |~ru × ~rv|dudv, the flux integral weights

each area element dS with the normal component of the vector field with ~F (~r(u, v) · ~n(~r(u, v)).We do not use this formula for computations because computing ~n gives additional work. We justdetermine the vectors ~F (~r(u, v)) and ~ru × ~rv and integrate its dot product over the domain.

1 Compute the flux of ~F (x, y, z) = 〈0, 1, z2〉 through the upper half sphere S parametrized by

~r(u, v) = 〈cos(u) sin(v), sin(u) sin(v), cos(v)〉 .

Solution. We have ~ru × ~rv = − sin(v)~r and ~F (~r(u, v)) = 〈0, 1, cos2(v)〉 so that∫

2π

0

∫ π

0

−〈0, 1, cos2(v)〉 · 〈cos(u) sin2(v), sin(u) sin2(v), cos(v) sin(v)〉 dudv .

The flux integral is∫2π0

∫ ππ/2− sin2(v) sin(u)−cos3(v) sin(v) dudv which is−

∫ ππ/2 cos

3 v sin(v) dv =

cos4(v)/4|π/20 = −1/4.

2 Calculate the flux of the vector field ~F (x, y, z) = 〈1, 2, 4z〉 through the paraboloid z =

x2 + y2 lying above the region x2 + y2 ≤ 1. Solution: We can parametrize the surface as~r(r, θ) = 〈r cos(θ), r sin(θ), r2〉 where ~rr ×~rθ = 〈−2r2 cos(θ),−2r2 sin(θ), r〉 and ~F (~r(u, v)) =

〈1, 2, 4r2〉. We get∫S~F · ~dS =

∫2π0

∫1

0(−2r2 cos(v)− 4r2 sin(v) + 4r3) drdθ = 2π.

3 Compute the flux of ~F (x, y, z) = 〈2, 3, 1〉 through the torus parameterized as ~r(u, v) =

〈(2 + cos(v)) cos(u), (2 + cos(v)) sin(u), sin(v)〉, where both u and v range from 0 to 2π.Solution. There is no computation is needed. Think about what the flux means.

4 Evaluate the flux integral∫ ∫

S〈0, 0, yz〉 ·~dS, where S is the surface with parametric equation

x = uv, y = u+ v, z = u− v on R : u2 + v2 ≤ 1. Solution: ~ru = 〈v, 1, 1〉, ~rv = 〈u, 1,−1〉 so

that ~ru × ~rv = 〈−2, u+ v,−u + v〉. The flux integral is∫∫

R〈0, 0, u2 − v2〉 · 〈−2, u + v,−u +

v〉 dudv =∫∫

Rv2u − u3 − v3 + u2v dudv which is best evaluated using polar coordinates:∫

1

0

∫2π0

r4(sin2(θ) cos(θ)− cos3(θ)− sin3(θ) + cos2(θ) sin(θ)) dθdr = 0.

5 Evaluate the flux integral∫∫

Scurl(F ) · ~dS for ~F (x, y, z) = 〈xy, yz, zx〉, where S is the part

of the paraboloid z = 4 − x2 − y2 that lies above the square [0, 1] × [0, 1] and has anupward orientation. Solution: curl(F ) = 〈−y,−z,−x〉. The parametrization ~r(u, v) =

〈u, v, 4−u2−v2〉 gives ru×rv = 〈2u, 2v, 1〉 and curl(F )(~r(u, v)) = 〈−v, u2+v2−4,−u〉. Theflux integral is

∫1

0

∫1

0〈−2uv+2v(u2+v2−4)−u〉 dvdu = −1/2+1/3+1/2−4−1/2 = −25/6.

6 What is the relation between the flux of the vector field ~F = ∇g/|∇g| through the surfaceS : {g = 1} with g(x, y, z) = x6+y4+2z8 and the surface area of S? Solution. The surface

area is equal to the flux because ~F · (~ru × ~rv) = |~ru × ~rv|.

7 Find the flux of the vector field ~G = ∇g × 〈0, 0, 1〉 through the surface S. Solution. Theflux is zero because the vector field is tangent to the surface.

8 Compute the flux of ~F (x, y, z) = 〈2, 3, 1〉 through the torus parameterized as ~r(u, v) =〈(2 + cos(v)) cos(u), (2 + cos(v)) sin(u), sin(v)〉, where both u and v range from 0 to 2π.

Solution. No computation is needed. Think about what the flux means.

9 You walk in the rain with velocity v along the x axis. The rain can be treated a fluid with

velocity ~F = 〈−v, 0, w〉, where w is the speed of the rain drops. Find the flux through yourfront and top if we assume you are a rectangular box [0, 1/2]× [0, 1]× [1/4, 1/3].

Solution. The flux through the surface S consisting of front and top is the amount of wateryou soak in per second. Parameterize the front by ~r(u, v) = 〈0, u, v〉 in the yz-plane. Your

front shirt and trousers catch in unit time the flux of the rain through the front surface.The head gets the rest. We know ~ru × ~rv = 〈0, 0, 1〉 for the top and ~ru × ~rv = 〈1, 0, 0〉 for

the front. The flux∫ ∫

S~F · ~dS is

∫1/4

0

∫1/3

0

〈−v, 0, w〉 · 〈0, 0, 1〉 dudv +∫

1/2

0

∫1

0

〈−v, 0, w〉 · 〈1, 0, 0〉 dudv = w/12 + v/2 .

If you walk for a fixed distance L and time T = L/v, the total flux is (L/v)(w/12 + v/2) =

L/2 + Lw/(2v). The part soaked up in the front depends only on L and not on the speeds.The flux caught by head and shoulder is less if you walk fast. So better run fast!


Lecture 32: Stokes theorem

The boundary of a surface is a curve oriented so thatthe surface is to the ”left” if the normal vector to thesurface is pointing ”up”. In other words, the velocityvector v, a vector w pointing towards the surface andthe normal vector n to the surface form a right handedcoordinate system.

Stokes theorem: Let S be a surface bounded by a curve C and ~F be a vector field.Then ∫ ∫

Scurl(~F ) · ~dS =

∫C

~F · ~dr .

1 Let ~F (x, y, z) = 〈−y, x, 0〉 and let S be the upper semi hemisphere, then curl(~F )(x, y, z) =〈0, 0, 2〉. The surface is parameterized by ~r(u, v) = 〈cos(u) sin(v), sin(u) sin(v), cos(v)〉 on

G = [0, 2π] × [0, π/2] and ~ru × ~rv = sin(v)~r(u, v) so that curl(~F )(x, y, z) · ~ru × ~rv =

cos(v) sin(v)2. The integral∫ 2π0

∫ π/20 sin(2v) dvdu = 2π.

The boundary C of S is parameterized by ~r(t) = 〈cos(t), sin(t), 0〉 so that ~dr = ~r ′(t)dt =

〈− sin(t), cos(t), 0〉 dt and ~F (~r(t)) ~r ′(t)dt = sin(t)2 + cos2(t) = 1. The line integral∫C~F · ~dr

along the boundary is 2π.

2 If S is a surface in the xy-plane and ~F = 〈P,Q, 0〉 has zero z component, then curl(~F ) =

〈0, 0, Qx − Py〉 and curl(~F ) · ~dS = Qx − Py dxdy. In this case, Stokes theorem can be

seen as a consequence of Green’s theorem. The vector field F induces a vector field on the

surface such that its 2D curl is the normal component of curl(F ). The reason is that thethird component Qx−Py of curl(~F )〈Ry −Qz, Pz −Rx, Qx−Py〉 is the two dimensional curl:~F (~r(u, v)) · 〈0, 0, 1〉 = Qx − Py. If C is the boundary of the surface, then

∫ ∫S~F (~r(u, v)) ·

〈0, 0, 1〉 dudv =∫C~F (~r(t)) ~r ′(t)dt.

3 Calculate the flux of the curl of ~F (x, y, z) = 〈−y, x, 0〉 through the surface parameterizedby ~r(u, v) = 〈cos(u) cos(v), sin(u) cos(v), cos2(v)+ cos(v) sin2(u+π/2)〉. Because the surface

has the same boundary as the upper half sphere, the integral is again 2π as in the aboveexample.

A surface is closed if it bounds a solid.

The flux of the curl of a vector field through a closed surface is zero.

4

The electric field E and the magnetic field B are linked by the

Maxwell equation curl( ~E) = −1cB. Take a closed wire C

which bounds a surface S and consider∫ ∫

S B · dS, the flux of

the magnetic field through S. The flux change is related witha voltage using Stokes theorem: d/dt

∫ ∫S B·dS =

∫ ∫S B·dS =∫ ∫

S −c curl( ~E) · ~dS = −c∫C~E ~dr = U , where U is the voltage

measured at the cut-up wire. The flux can be changed by

turn around a magnet around the wire or the wire inside themagnet, we get an electric voltage. Stokes theorem explains

why we can generate electricity from motion.

George Gabriel Stokes Andre Marie Ampere

5 Find∫C~F · ~dr, where ~F (x, y, z) = 〈x2y, x3/3, xy〉 and C is the curve of intersection of

the hyperbolic paraboloid z = y2 − x2 and the cylinder x2 + y2 = 1, oriented counter-

clockwise as viewed from above. Solution. The curl of F is curl(F ) = (x,−y, 0). We canparametrize the hyperbolic paraboloid as ~r(u, v) = (u cos(v), u sin(v),−u2 cos(2v)). ~ru×~rv =

〈2u2 cos(v),−2u2 sin(v), u〉. ~F (~r(u, v)) = 〈u cos(v),−u sin(v), 0〉. ~F (~r(u, v)) · ~ru × ~rv) = 2u3.The integral is

∫ 10

∫ 2π0 −2r3 dθdr = π.

6 Evaluate the flux of ~F (x, y, z) = 〈xey2

z3 + 2xyzex2+z, x + z2ex

2+z, yex2+z + zex〉 through

the part S of the ellipsoid x2 + y2/4 + (z + 1)2 = 2, z > 0 oriented so that the normal

vector points upwards. Solution. Stokes theorem assures that the flux integral is equalto the line integral along the boundary of the surface. The boundary is the ellipse ~r(t) =

〈cos(t), 2 sin(t)〉, 0 ≤ t ≤ 2π. The vector field on the xy-plane z = 0 is

~F (x, y, 0) = 〈0, x, yex2

〉 .

To compute the line integral of this vector field along the boundary curve, compute ~r′(t) =

〈− sin(t), 2 cos(t), 0〉 and ~F (~r(t)) = 〈0, cos(t), 2 sin(t)esin2(t)〉. The dot product is the power

2 cos2(t). Integrating this over [0, 2π] gives 2π.


Lecture 32: Stokes theorem

To understand Stokes, like with anything in mathematics, you have first to make sure that alldefinitions are clear. This is already quite tough for Stokes theorem, because it involves a lot ofconcepts:

• parametrized curves: the boundary of a surface is a curve

• parametrized surfaces: the surface is usually given as such

• line integral∫I~F (~r(t)) · ~r ′(t) dt

• flux integral∫G~F (~r(u, v) · (~ru × ~rv) dudv

• the dot and cross product appear. In the flux integral you can see the triple scalar product

• the result relates a single variable integral with a double integral

Here is the theorem:

Stokes theorem: Let S be a surface bounded by a curve C and ~F be a vector field.Then ∫ ∫

Scurl(~F ) · ~dS =

∫C

~F · ~dr .

Stokes theorem has been assigned as an exam problem in 1854 by George Stokes. George Maxwell,(the one from the Maxwell equations) took the exam. It clearly influenced him. See the exhibiton the course website.

Stokes theorem is important because:

• It is a crucial prototype of a theorem.

• It crowns multivariable calculus reinforcing older concepts. 1

• When understanding it, you demonstrate ability of abstract thought.

• It explains concepts in physics, especially fluid dynamics and electromagnetism.

• It explains what curl is.

After knowing the definitions which go into the theorem and the statement of the theorem youcan use the theorem as a tool. It is at first not so important to worry about the meaning of thingstoo much but to learn how to use the theorem as a tool, for example to solve complicated lineintegrals or to avoid complex flux integrals. Here are two very typical examples how to use thetheorem:

1As a general principle, it is always a good idea to learn more about a subject than you actually need.

1 Problem A: Find the flux of the curl of ~F (x, y, z) = 〈−y+sin(x)z, x, z3〉 through the surface

x2 + y2/4 + z2 + z4x2 = 1, z > 0. Solution. This is a problem where the flux integral wouldbe tough to compute. We would have to parametrize the surface and if we would succeed

doing that compute the integral. Much easier is to use Stokes theorem and to compute theline integral of ~F along the boundary of the surface which is an ellipse in the xy plane.

2 Problem B: Find the line integral∫C~F ~dr, where C is the circle of radius 3 in the xz-plane

oriented counter clockwise when looking from the point (0, 1, 0) onto the plane and where~F is the vector field

~F (x, y, z) = 〈2x2z + x5, cos(ey),−2xz2 + sin(sin(z)〉 .

Use a convenient surface S which has C as a boundary. Solution The line integral can not

be computed directly. The curl of ~F is 〈0, 2x2 + 2z2, 0〉. It is∫3

0

∫2π0

2r3 dθdr. The answer is81π.

Here are two conceptual problems:

3 If S is the surface x6 + y6 + z6 = 1 and assume ~F is a smooth vector field. Explain why∫∫Scurl(

~F ) · ~dS = 0. Solution. The flux of curl(F ) through a closed surface is zero by Stokestheorem and the fact that the surface does not have a boundary.

One can see this also by cutting the surface in two pieces and apply Stokes to both pieces.

4 Assume you look at the surface S which is the boundary of a doughnut. You cut the

doughnut in two pieces. Verify that the flux of the curl through one part of the surfaceis the opposite than the flux through the other. Solution. One can understand this with

Stokes theorem. The flux of the first part is equal to the line integral along the boundary.The flux through the second is the line integral along the same boundary but in the opposite

direction.

5 Why is simply connectedness of the region under consideration needed to verify that if a

vector field has zero curl then it is a gradient field? This problem appears in the movie ”Abeautiful mind”.

Solution. Given a closed loop C in space. Simply connectedness means that one can pulltogether the loop to a point. This pulling together defines a surface S to which the curve

C is a boundary. The flux of the curl of F through S is zero. By Stokes theorem, the line

integral is zero. Now F has the closed loop property and is therefore a gradient field. If theregion is not simply connected like the complement of the z-axes, then we can find vector

fields F which have zero curl everywhere in the region but which are not gradient fields. Anexample is

~F (x, y, z) = 〈−y/(x2 + y2), x/(x2 + y2), 0〉 .


Lecture 33: Divergence theorem

There are three integral theorems in three dimensions. We have seen already the fundamentaltheorem of line integrals and Stokes theorem. Here is the divergence theorem, which completesthe list of integral theorems in three dimensions:

Divergence Theorem. Let E be a solid with boundary surface S oriented so thatthe normal vector points outside. Let ~F be a vector field. Then

∫ ∫ ∫Ediv(~F ) dV =

∫ ∫S

~F · dS .

To prove this, one can look at a small box [x, x + dx] ×

[y, y+dy]× [z, z+dz]. The flux of ~F = 〈P,Q,R〉 through

the faces perpendicular to the x-axes is [~F (x + dx, y, z) ·

〈1, 0, 0〉 + ~F (x, y, z) · 〈−1, 0, 0〉]dydz = P (x + dx, y, z) −P (x, y, z) = Px dxdydz. Similarly, the flux through they-boundaries is Py dydxdz and the flux through the two z-boundaries is Pz dzdxdy. The total flux through the facesof the cube is (Px + Py + Pz) dxdydz = div(~F ) dxdydz.A general solid can be approximated as a union of smallcubes. The sum of the fluxes through all the cubes con-sists now of the flux through all faces without neigh-boring faces. and fluxes through adjacent sides can-cel. The sum of all the fluxes of the cubes is the fluxthrough the boundary of the union. The sum of allthe div(~F ) dxdydz is a Riemann sum approximation for

the integral∫ ∫ ∫

G div(~F ) dxdydz. In the limit, wheredx, dy, dz goes to zero, we obtain the divergence theorem.

The theorem explains what divergence means. If we average the divergence over a small cube isequal the flux of the field through the boundary of the cube. If this is positive, then more fieldexists the cube than entering the cube. There is field ”generated” inside. The divergence measuresthe expansion of the field.

1 Let ~F (x, y, z) = 〈x, y, z〉 and let S be sphere. The divergence of ~F is the constant function

div(~F ) = 3 and∫ ∫ ∫

G div(~F ) dV = 3 · 4π/3 = 4π. The flux through the boundary is∫ ∫S ~r · ~ru × ~rv dudv =

∫ ∫S |~r(u, v)|

2 sin(v) dudv =∫ π0

∫2π0

sin(v) dudv = 4π also. We see that

the divergence theorem allows us to compute the area of the sphere from the volume of theenclosed ball or compute the volume from the surface area.

2 What is the flux of the vector field ~F (x, y, z) = 〈2x, 3z2 + y, sin(x)〉 through the solid

G = [0, 3]× [0, 3]× [0, 3] \ ([0, 3]× [1, 2]× [1, 2] ∪ [1, 2]× [0, 3]× [1, 2] ∪ [0, 3]× [0, 3]× [1, 2])which is a cube where three perpendicular cubic holes have been removed? Solution: Use

the divergence theorem: div(~F ) = 2 and so∫ ∫ ∫

G div(~F ) dV = 2∫ ∫ ∫

G dV = 2Vol(G) =2(27 − 7) = 40. Note that the flux integral here would be over a complicated surface over

dozens of rectangular planar regions.

3 Find the flux of curl(F ) through a torus if ~F = 〈yz2, z + sin(x) + y, cos(x)〉 and the torus

has the parametrization

~r(θ, φ) = 〈(2 + cos(φ)) cos(θ), (2 + cos(φ)) sin(θ), sin(φ)〉 .

Solution: The answer is 0 because the divergence of curl(F ) is zero. By the divergence

theorem, the flux is zero.

4 Similarly as Green’s theorem allowed to calculate the area of a region by passing along the

boundary, the volume of a region can be computed as a flux integral: Take for example thevector field ~F (x, y, z) = 〈x, 0, 0〉 which has divergence 1. The flux of this vector field through

the boundary of a solid region is equal to the volume of the solid:∫ ∫

δG〈x, 0, 0〉·~dS = Vol(G).

5

How heavy are we, at distance r from the center of the

earth? Solution: The law of gravity can be formulatedas div(~F ) = 4πρ, where ρ is the mass density. We as-

sume that the earth is a ball of radius R. By rotationalsymmetry, the gravitational force is normal to the sur-

face: ~F (~x) = ~F (r)~x/||~x||. The flux of ~F through a ballof radius r is

∫ ∫Sr

~F (x) · ~dS = 4πr2 ~F (r). By the di-

vergence theorem, this is 4πMr = 4π∫ ∫ ∫

Br

ρ(x) dV ,

where Mr is the mass of the material inside Sr. We have(4π)2ρr3/3 = 4πr2 ~F (r) for r < R and (4π)2ρR3/3 =

4πr2 ~F (r) for r ≥ R. Inside the earth, the gravitationalforce ~F (r) = 4πρr/3. Outside the earth, it satisfies~F (r) = M/r2 with M = 4πR3ρ/3.

0.5 1.0 1.5 2.0

0.2

0.4

0.6

0.8

1.0


Lecture 34: Overview

All integral theorems are incarnations of thefundamental theorem of multivariable

Calculus

∫G

dF =∫δG

F

where dF is a derivative of F and δG is theboundary of G.

1

1FTC−→ 1

1FTL−→ 2

Green−→ 1

1FTL−→ 3

Stokes−→ 3

Gauss−→ 1

Fundamental theorem of line integrals: If C is a curve with boundary {A,B}and f is a function, then

∫C

∇f · ~dr = f(B)− f(A)

Remarks.

1) For closed curves, the line integral∫C∇f · ~dr is zero.

2) Gradient fields are path independent: if ~F = ∇f , then the line integral between twopoints P and Q does not depend on the path connecting the two points.3) The theorem justifies the name conservative for gradient vector fields.4) The term ”potential” was coined by George Green who lived from 1793-1841.

1 Let f(x, y, z) = x2+ y4+ z. Find the line integral of the vector field ~F (x, y, z) = ∇f(x, y, z)along the path ~r(t) = 〈cos(5t), sin(2t), t2〉 from t = 0 to t = 2π.

Solution. ~r(0) = 〈1, 0, 0〉 and ~r(2π) = 〈1, 0, 4π2〉 and f(~r(0)) = 1 and f(~r(2π)) = 1 + 4π2.The fundamental theorem of line integral gives

∫C∇f ~dr = f(r(2π))− f(r(0)) = 4π2.

Green’s theorem. If R is a region with boundary C and ~F is a vector field, then

∫ ∫R

curl(~F ) dxdy =∫C

~F · ~dr .

Remarks.1) The curve is oriented in such a way that the region is to the left.2) The boundary of the curve can consist of piecewise smooth pieces.3) If C : t 7→ ~r(t) = 〈x(t), y(t)〉, the line integral is

∫b

a〈P (x(t), y(t)), Q(x(t), y(t))〉 · 〈x′(t), y′(t)〉 dt.

4) Green’s theorem was found by George Green (1793-1841) in 1827 and by Mikhail Ostrogradski(1801-1862).

5) If curl(~F ) = 0 everywhere in the plane, then the field is a gradient field.

7) ~F (x, y) = 〈0, x〉 we get an area formula.

2 Find the line integral of the vector field ~F (x, y) = 〈x4 + sin(x) + y, x+ y3〉 along the path

~r(t) = 〈cos(t), 5 sin(t) + log(1 + sin(t))〉, where t runs from t = 0 to t = π.Solution. curl(~F ) = 0 implies that the line integral depends only on the end points

(0, 1), (0,−1) of the path. Take the simpler path ~r(t) = 〈−t, 0〉,−1 ≤ t ≤ 1, which has veloc-ity ~r ′(t) = 〈−1, 0〉. The line integral is

∫1

−1〈t4−sin(t),−t〉·〈−1, 0〉 dt = −t5/5|1

−1= −2/5.

Stokes theorem. If S is a surface with boundary C and ~F is a vector field, then

∫ ∫S

curl(~F ) · dS =∫C

~F · ~dr .

Remarks.1) Stokes theorem allows to derive Greens theorem: if ~F is z-independent and the surface S iscontained in the xy-plane, one obtains the result of Green.2) The orientation of C is such that if you walk along C and have your head in the direction ofthe normal vector ~ru × ~rv, then the surface to your left.3) Stokes theorem was found by Andre Ampere (1775-1836) in 1825 and rediscovered by GeorgeStokes (1819-1903).

4) The flux of the curl of ~F only depends on the boundary of S.5) The flux of the curl through a closed surface like the sphere is zero because the boundary isempty.

3 Compute the line integral of ~F (x, y, z) = 〈x3+xy, y, z〉 along the polygonal path C connectingthe points (0, 0, 0), (2, 0, 0), (2, 1, 0), (0, 1, 0).

Solution. The path C bounds a surface S : ~r(u, v) = 〈u, v, 0〉 parameterized by R = [0, 2]×[0, 1]. By Stokes theorem, the line integral is equal to the flux of curl(~F )(x, y, z) = 〈0, 0,−x〉

through S. The normal vector of S is ~ru × ~rv = 〈1, 0, 0〉 × 〈0, 1, 0〉 = 〈0, 0, 1〉 so that∫ ∫Scurl(~F ) · ~dS =

∫2

0

∫1

0〈0, 0,−u〉 · 〈0, 0, 1〉 dudv =

∫2

0

∫1

0−u dudv = −2.

Divergence theorem: If S is the boundary of a region E in space and ~F is avector field, then ∫ ∫ ∫

B

div(~F ) dV =∫ ∫

S

~F · ~dS .

Remarks.1) The divergence theorem is also called Gauss theorem.2) It can be helpful to determine the flux of vector fields through surfaces.3) It was discovered in 1764 by Joseph Louis Lagrange (1736-1813), later it was rediscovered byCarl Friedrich Gauss (1777-1855) and by George Green.

4) For divergence free vector fields ~F , the flux through a closed surface is zero. Such fields ~F arealso called incompressible or source free.

4 Compute the flux of the vector field ~F (x, y, z) = 〈−x, y, z2〉 through the boundary S of therectangular box [0, 3]× [−1, 2]× [1, 2].

Solution. By Gauss theorem, the flux is equal to the triple integral of div(F ) = 2z overthe box:

∫3

0

∫2

−1

∫2

12z dxdydz = (3− 0)(2− (−1))(4− 1) = 27.

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